Ω-Theory: Mathematics with Infinite and Infinitesimal Numbers - SELP

UNIVERSITÀ DEGLI STUDI DI PAVIA
FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI
CORSO DI LAUREA IN MATEMATICA
Ω-Theory:
Mathematics with Infinite and
Infinitesimal Numbers
Relatore
Chiar.mo Prof. Vieri Benci
Correlatore
Chiar.mo Prof. Mauro Di Nasso
Correlatore
Chiar.mo Prof. Enrico Vitali
Anno Accademico 2010/2011
Tesi di laurea di
Emanuele Bottazzi

To Eleonora

Contents
1 Ω-Calculus 1
1.1 Infinitesimal and infinite numbers . . . . . . . . . . . . . . . . . . . . 2
1.2 Superreal Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 The Axioms of Ω-Calculus . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 First properties of Ω-limit . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 The Transfer Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Numerosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Extensions of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.8 Extensions of functions . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.9 Towards a model for Ω-Calculus . . . . . . . . . . . . . . . . . . . . . 27
1.10 Filters and ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.11 A model for Ω-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.12 Reasoning in the model . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Selected topics of Ω-Calculus 43
2.1 Convergence and continuity . . . . . . . . . . . . . . . . . . . . . . . 44
2.2 Topology by Ω-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Integrals by Ω-Calculus 54
3.1 Hyperfinite sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Ω-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Ω-integral and Riemann integral . . . . . . . . . . . . . . . . . . . . . 59
3.4 Numerosity and Lebesgue measure . . . . . . . . . . . . . . . . . . . 61
4 Ω-Theory 65
4.1 Foundational framework . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 The Axioms of Ω-Theory . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Generalizing concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4 Transfer Principle by Ω-Theory . . . . . . . . . . . . . . . . . . . . . 70
4.5 A model for Ω-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 The formal version of Transfer Principle 79
5.1 Logic formalism for set theory . . . . . . . . . . . . . . . . . . . . . . 79
5.2 Transfer Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
v

6 An application: Non-Archimedean Probability 87
6.1 Kolmogorov’s axioms of probability . . . . . . . . . . . . . . . . . . . 87
6.2 Non-Archimedean probability . . . . . . . . . . . . . . . . . . . . . . 89
6.3 First consequences of NAP axioms . . . . . . . . . . . . . . . . . . . 91
6.4 Fair lotteries by NAP . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Infinite sequence of coin tosses by NAP . . . . . . . . . . . . . . . . . 97
7 Further ideas 99
7.1 The Cartesian Product Axiom for Numerosity . . . . . . . . . . . . . 99
7.2 Dimension by Ω-Theory . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Numerosity and Lebesgue measure . . . . . . . . . . . . . . . . . . . 103
7.4 Functional Analysis by Ω-Theory . . . . . . . . . . . . . . . . . . . . 104
7.5 R ∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.6 Ω-integral of hyperreal functions . . . . . . . . . . . . . . . . . . . . . 106
7.7 The Dirac distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.8 The natural extension of numerosity . . . . . . . . . . . . . . . . . . 108
vi

Introduction
Borknagar
La mia vita in quest’ultimo anno è
stata un susseguirsi di peripezie.
[...] Ma sono contento di tutto
quello che ho fatto, del capitale di
esperienze che ho accumulato, anzi
avrei voluto fare di più.
Italo Calvino
We believe that there are many ways to do mathematics, and this belief is
motivated by a number of examples. One of the simplest but nonetheless
significative is given by the field of complex numbers: historically, it has
been presented as the splitting field of the polynomials with real coefficients, but in
the language of modern algebra this field is defined as the quotient of the polynomial
ring R[x] by the principal ideal generated by x2 + 1. Of course, it can be proved
that the two definitions are equivalent, but each of them has its own role, both in
the history and in the practice of mathematics.
We find also very interesting that a theory that has been given many different
formalizations during the centuries is the Euclidean geometry. In particular,
during the XX century, many mathematicians have given their own axiomatics for
Euclidean geometry: here, we want to focus on Hilbert’s presentation introduced
in the Grundlagen der Geometrie and on Prodi and Choquet formulations. The
intent of Hilbert’s axiomatic is clear: to give a correct and elegant foundation of Euclidean
geometry with the precision of “modern” mathematics. In particular, one of
the characteristics of this approach is that, for the working mathematician, nothing
more than the axioms of the theory is needed to develop all Euclidean geometry.
On the other hand, both Prodi and Choquet aim to give an axiomatic system for
Euclidean geometry simple enough to be taught to high school students. Hence,
their axiomatics are less tecnical than Hilbert’s one, and rely heavily on the structure
of the real numbers. In this case, it’s clear that some previous mathematical
knowledge of the real numbers is necessary for the development of the theory.
vii

INTRODUCTION
It’s a well known fact that even calculus, whose classical approach based on
the “ε-δ” definition of limit has been formulated in the XIX century, can be given
a different formalization based on the concepts of infinite and infinitesimal numbers.
The resulting theory, usually called Nonstandard Analysis, has proved to give
means to develop calculus in a way closer to the mathematical intuition than the
“standard” one, but without sacrificing precision or mathematical rigor. While we
do agree that intuition shouldn’t be used as a valid inference tool of mathematics,
we are equally convinced that many nonstandard definitions linked to the topics of
convergence and continuity could help math students (and even some teachers) to
better understand the underlying notions without hiding them under a layer of cryptic
formalism. In this regard, it’s also been argued that the “quantifier complexity”
of some nonstandard definitions is indeed reduced with respect to standard ones.
The main weakpoint of Nonstandard Analysis is that the formalism of Robinson’s
original presentation appeared too technical to many, and not directly usable by
those mathematicians without a good background in logic. Over the last forty
years, many attempts have been made in order to simplify the foundational matters
and popularize nonstandard analysis by means of “easy to grasp” presentations.
Following those attempts, the mathematical theories of Ω-Calculus and Ω-Theory
presented in this paper are two simplified version of Nonstandard Analysis expressed
in the everyday language of mathematics, in a way to still allow for a complete and
rigorous treatment of all the basics of calculus. An important and fascinating feature
of those theories lies in the fact that, in contrast with the original formulation of
Nonstandard Analysis, their presentation only assumes a few basic notions from
algebra, so that they need no more prerequisites than the usual Calculus does. For
those reasons, we do believe that Ω-Calculus and its extension given by Ω-Theory
are two beautiful and relatively simple theories which could potentially bring many
mathematicians, especially those who have no background in logic, closer to the
methods of Nonstandard Analysis.
Since, in analogy to what happens for Nonstandard Analysis, Ω-Calculus and Ω-
Theory are both grounded on the use of infinite and infinitesimal numbers, the first
chapter of this paper is dedicated to the introduction of the basic definitions of those
concepts. After that, we will introduce the axiomatical theory of Ω-Calculus and the
fundamental operation of “Ω-limit”, and we will study in detail their properties and
some of the more significative applications in extending real functions to particular
hyperreal sets. Once the reader is acquainted with those results, there shouldn’t be
more difficulties in following the rest of the paper. It’s interesting to remark that, in
order to give the axioms of Ω-Calculus and to settle their most useful consequences,
only a little knowledge of algebra is required.
The last three sections of the first chapter are dedicated to the explicit construction
of an algebraic model for Ω-Calculus: the main goal is to show that the
axioms of Ω-Calculus are consistent but, in the course of this paper, we will see that
the algebraic models of Ω-Calculus can be also used to give a nice interpretation
to the operation of Ω-limit and are very useful in view of some applications, most
notably to probability. We signal that some notions about rings, ideals and fields are
necessary to fully understand the model building process, but the necessary ideas
don’t exceed what is usually taught during an introductive course in algebra. Some
viii

INTRODUCTION
notions of set theory are also required, but we will give a brief presentation of the
main concepts and results we will use.
Chapters 2 and 3 are dedicated to show the Ω-Calculus “in action”. During the
first part of Chapter 2, we will show the genesis of the concept of convergence for
real sequences and for real functions; in the second part, we will focus on how the
usual topology of the real line can be interpreted by using Ω-Calculus, and how
this new interpretation seems to simplify the proofs of some famous theorems, the
most important of them being the Banach-Caccioppoli Fixed Point Theorem. In the
course of Chapter 3, we will introduce an interesting notion of Ω-integral that makes
sense for all real functions, and then we will show its relations between Riemann
and Lebesgue integral.
The theory introduced so far will be generalized in Chapter 4, where the concepts
introduced in Chapter 1 will be extended to functions with arbitrary range. The
resulting theory, that will be called Ω-Theory, will be interesting on its own, but
can also be applied in a number of situations. One of the possible applications
of Ω-Theory will be given in Chapter 6, where we will present a new approach to
probability that overcomes some of the weakpoints of the classical Kolmogorovian
approach, namely the fact that the union of elementary events is not always an event
and the vague interpretation of probability measures for which there exist nonempty
events with zero probability. This chapter will be rich of examples, in order to focus
on some “tangible” results and not only on the abstract theory.
The possible applications of Ω-Theory to everyday mathematics seems to be very
broad, and surely exceed the scope of this paper. In the conclusive chapter, we have
selected a few of the most interesting ideas that couldn’t be properly developed and,
for each of them, we have given a brief presentation. We hope that, one day, many
of those open questions will be given an answer.
Chapter 5 is planned as a “logic intermission”, where we will show that the Ω-
Theory proves a very important logic theorem, namely the Transfer Principle. This is
the only chapter where a little logic knowledge is required to fully master the results.
We remark that the main instances of Transfer that will be used in this paper can
be directly shown without any logic knowledge, so our claim that Ω-Theory can be
used as a more advanced theory to introduce those mathematicians who have little
or no background in logic close to the methods of Nonstandard Analysis still holds.
A few words about notation
We believe that uniformity in
graphical terminology will never be
attained, and is not necessarely
desirable.
Frank Harary
If A and B are two sets, we will use the symbol A ⊆ B when A is a subset of B
and we admit the possibility that A = B, or we don’t know yet if A = B is true or
ix

INTRODUCTION
not. We will write A ⊂ B if A is a subset of B and we don’t admit the possibility
that A = B, or we already know that A = B.
If A and B are sets, we will denote by A × B the cartesian product of A and B.
For a matter of commodity, if A is a set and n ∈ N, in the last chapter of this paper
we will sometimes denote by An the cartesian product A
× .
. . × A
.
n times
We will denote by | · | the cardinality of a set.
If A is a set, we will denote by P(A) its powerset. Moreover, we define once and
for all the set Pfin(A) = {λ ⊆ A : 0 < |λ| < +∞}. Clearly, Pfin(A) ⊂ P(A), and
the two differ only for the empty set if and only if |A| is finite.
If A ⊆ Ω, by Ac we will denote the set Ω \ A. We will use the same notation
even in other cases, but it will always be clear what is the set B that has the same
role Ω had in the previous case.
For all sets A and B, for all b ∈ B we will denote (with abuse of notation) by cb
the function cb : A → B defined by cb(a) = b for all a ∈ A. We will denote by χA
the function χA : A → {0, 1} defined by
1 if x ∈ A
χA(x) =
0 if x ∈ A
and we will call it the characteristic function of A.
We will denote by N, Z, Q and R the sets of, respectively, natural numbers,
integer numbers, rational numbers and real numbers. For a matter of commodity,
we will agree that N = {1, 2, 3, . . .}, so that 0 ∈ N.
If F is an ordered field and a, b ∈ F, we will write [a, b)F = {x ∈ F : a ≤ x < b}.
We will use the notations [a, b]F, (a, b)F and (a, b]F accordingly. When there will be
littke risk of misunderstanding, we will omit the indication of the field F. If x ∈ F,
we will denote by |x| its absolute value:
x if x ≥ 0
|x| =
−x if x < 0
x

Chapter 1
Ω-Calculus
Mathematics is not a careful march
down a well-cleared highway, but a
journey into a strange wilderness,
where the explorers often get lost.
W. S. Anglin
The intuitive notions of “infinitely small” and “infinitely large” quantities
are central in the development of differential calculus. The current “ε-δ
approach”, as elaborated by Weierstrass in the second half of the XIX century,
incorporates those notions in an indirect manner that is grounded on the notion
of limit. This, however, is not the only way to give a mathematical formalization
of those ideas, as most notably shown by Abraham Robinson with the introduction
of Nonstandard Analysis. Once given a mathematical theory for infinitesimal and
infinite numbers, it’s only natural to wonder if calculus can be grounded on those
notions, instead of on the use of limits. It’s a well known fact that this is indeed the
case, and the Ω-Calculus that will be presented in this chapter is a new elementary
method for developing calculus in such a way.
In the first two sections of this chapter, we will formally define what infinitesimal
and infinite numbers are and how they behave under the basic algebraics operations.
In those sections, we will give some fundamental definitions and results that will be
used through all the paper, and most notably we will introduce the concept of
superreal fields and of standard part of a superreal number.
In Section 1.3, we will give the axioms of Ω-Calculus, that will rule the existence
and the behaviour of a new kind of “limit”, that we will call “Ω-limit”. This “Ωlimit”
will allow a construction of a particular superreal field R∗ that, as we will see
Chapters 2 and 3, will be the natural setting where basic calculus can be developed.
The central part of this chapter is dedicated to show the main properties of
the Ω-limit and how to use it to introduce some interesting mathematical concepts,
namely numerosity and the natural and hyperfinite extension of sets and functions.
The most notable characteristic of those extension is the fact that they preserve all
“elementary properties” of the mathematical entities involved, in a way that will be
precised in Section 1.5.
1

CHAPTER 1. Ω-CALCULUS
Finally, in Sections 1.9, 1.10 and 1.11, we will address the problem of the consistency
of the axioms of Ω-Calculus, and we will show that they admit an algebraic
model.
In the last section of this chapter, we will show some examples of how the model
can be used to infer some properties of the theory that are not ruled out by the
axioms.
1.1 Infinitesimal and infinite numbers
Infinity is not a big number.
John D. Barrow
In our approach to calculus, we are interested in number fields containing infinitesimal
and infinite numbers. Before we define them in a rigorous way, we need
to recall some basic notions about ordered fields.
Definition 1.1.1. An ordered field is a couple (F, F + ) where F is a field, and F + ⊂ F
satisfies:
• for all x, y ∈ F + , then x + y and xy ∈ F;
• F = −F + ∪ {0} ∪ F + , where −F + = {−x : x ∈ F + }, and the union is disjoint.
Then, we can define an order on F by setting
x < y ⇐⇒ y − x ∈ F +
We say that the elements of F + are positive, and the elements of −F + are negative.
Following the common practice, we identify each positive n ∈ N with the corresponding
iterated sum of the neutral element 1 ∈ F:
n = 1 + . . . + 1
n times
Moreover, by considering opposites and reciprocals, we can suppose Q ⊆ F.
Now, we want to introduce the concept of infinitesimal numbers in a formal way.
The basic idea of infinitesimal numbers is that they are numbers whose absolute value
is “smaller than every other (natural) number”. Other than that, we want to be
able to perform field operations (sum, product, opposite and inverse) on infinitesimal
numbers, so we’ll require that infinitesimal numbers are contained in a field F. Since
the informal definition of infinitesimal numbers involves ordering, we require also
that F is an ordered field. Putting everything together, we can rewrite the intuitive
idea of infinitesimal number in the following way:
Definition 1.1.2. Let F be an ordered field. A number ξ ∈ F is called infinitesimal
if, for all n ∈ N, |ξ| < 1/n.
2

1.1. INFINITESIMAL AND INFINITE NUMBERS
Is this definition satisfied by some numbers? Let’s think about the field of rational
numbers and the field of real numbers: in those fields, there exists exactly one
infinitesimal number: the number 0.
Up to now, we don’t know if there exist some number fields containing infinitesimal
numbers different from 0 but, in section 1.11, we will give an explicit construction
of one of such fields. This result allows us to study ordered fields F ⊃ R that contains
nonzero infinitesimal numbers, knowing in advance that we’re not reasoning
about nonexistant objects.
The notion of infinitesimal numbers allows a classification of numbers according
to their “size”.
Definition 1.1.3. Let ξ ∈ F. We say that:
• ξ is infinitesimal if for all n ∈ N, |ξ| < 1
n ;
• ξ is finite if there exists n ∈ N such as |ξ| < n;
• ξ is infinite if, for all n ∈ N, |ξ| > n (equivalently, if ξ is not finite).
It’s easily seen that the inverse of a nonzero infinitesimal number is infinite, and
the inverse of an infinite number is infinitesimal. Clearly, all infinitesimal numbers
are finite. However, we remark that in any non-Archimedean field there are plenty of
finite numbers that are neither infinitesimal nor rational. In fact, for every non-zero
q ∈ Q and for every nonzero infinitesimal ξ, the number q + ξ ∈ Q is finite and
non-infinitesimal.
In the next proposition, we itemize a list of simple algebraic properties that
match the intuition of “small”, “finite”, and “infinite” quantities.
Proposition 1.1.4. 1. If ξ and ζ are finite, then ξ + ζ and ξ · ζ are finite;
2. If ξ and ζ are infinitesimal, then ξ + ζ and ξ · ζ are infinitesimal;
3. If ξ is infinitesimal and ζ is finite, then ξ · ζ is infinitesimal;
4. If ξ is infinite and ζ is not infinitesimal, then ξ · ζ is infinite;
5. If ξ = 0 is infinitesimal and ζ is not infinitesimal, ζ/ξ is infinite;
6. If ξ is infinite and ζ = 0 is finite, then ζ/ξ is infinitesimal.
Proof. All the proofs are elementary, and therefore left to the interested reader as
exercises. As an example, we will give the proof of part 5. We already know that, if
ξ = 0 is infinitesimal, 1/ξ is infinite. By definition of infinite number, we have that,
for all n ∈ N, |1/ξ| > n. Now, since ζ is not infinitesimal, there exists k ∈ N such
that |ζ| > 1/k. We can conclude |ζ/ξ| > n/k for all n ∈ N and this implies that ζ/ξ
is infinite.
Note that a non-Archimedean field cannot be complete. For instance, the set of
all infinitesimal numbers is upper bounded, but it has no least upper bound (if x is
an upper bound for this set, then also x/2 < x is).
The request that a field F contains infinitesimal number appears to be a fair
simple one, but it’s deeply connected to some other basic properties of the field, as
we’ll see in the next Proposition.
3

CHAPTER 1. Ω-CALCULUS
Proposition 1.1.5. An ordered field is Archimedean if and only if its only infinitesimal
number is 0.
Before we give the proof of this proposition, we recall that an ordered field is
Archimedean if the following property holds:
∀x > 0 ∀y > 0 ∃n ∈ N : ny > x
or, in other words, the iterated sums of any positive element y surpass any given
number x.
Proof of Proposition 1.1.5. Let ξ be an infinitesimal number. Without loss of generality,
we can suppose ξ > 0 (otherwise, choose −ξ). Since we have, by definition,
ξ < 1/n for all n ∈ N, if we consider the inverses we have n < 1/ξ for all n ∈ N or,
in other words, the field is not Archimedean.
Now, suppose that the field is not Archimedean. Then, by definition, we have
an element ξ that is a counterexample for the Archimedean property, that is n < ξ
for all n ∈ N. Then 1/ξ is a nonzero infinitesimal number, since 0 < 1/ξ < 1/n for
all n ∈ N.
Proposition 1.1.5 gives a very interesting characterization of non-Archimedean
fields, and shows that fields containing infinitesimal numbers don’t satisfy some
common properties we are used to take for granted.
1.2 Superreal Fields
In the mathematical practice and especially in analysis, the most used extension of
the real field is given by the complex numbers C. Instead, we want to concentrate
on those extensions of R that are still ordered fields.
Definition 1.2.1. A superreal field is an ordered field F that properly extends R
(e.g. R ⊂ F, R + ⊂ F + , and the inclusions are strict).
It turns out that any superreal field contains infinitesimal and infinite numbers,
as shown in the next Theorem.
Theorem 1.2.2. Any superreal field F is non-Archimedean.
Proof. Thanks to Proposition 1.1.5, we only need to show that F contains an infinite
or infinitesimal number. Choose ξ ∈ F\R. Without loss of generality, we can suppose
ξ > 0. If ξ > n for all n ∈ N, there’s nothing to prove. Let’s suppose this is not the
case, and 0 < ξ < n for some n ∈ N. Now, consider the set
X = {x ∈ R : ξ < x}
of the real upper bounds of ξ. By hypothesis, X is nonempty and bounded below.
Using the completeness property of R, the element r = inf X is a well defined real
number. We claim that ζ = ξ − r is an infinitesimal number.
To prove our claim, let’s pick n ∈ N. By the properties of least upper bound, we
have that ξ ≥ r − 1/n, so that ζ ≥ −1/n, and ξ < r + 1/n, so that ζ < 1/n. Since
both inequalities hold for all n ∈ N, we conclude that ζ is a nonzero infinitesimal
number.
4

1.2. SUPERREAL FIELDS
Thanks to infinitesimal numbers, in the superreal fields we can formalize a new
notion of “closeness”.
Definition 1.2.3. We say that two numbers ξ and ζ are infinitely close if ξ − ζ is
infinitesimal. In this case, we will write ξ ∼ ζ.
It’s easy to see that the relation ∼ if infinite closeness is an equivalence relation.
By using this relation, we can give an idea of the order-structure of superreal
fields. This, in turn, will allow a canonical representation of the finite numbers.
Theorem 1.2.4. Every finite number ξ is infinitely close to a unique real number
r ∼ ξ, called the shadow or the standard part of ξ. We will write r = sh(ξ).
Proof. The proof is similar to that of Theorem 1.2.2: let’s consider the set of the
real upper bounds of ξ
X = {x ∈ R : ξ < x}
and let r = inf X. We have seen in the proof of Theorem 1.2.2 that ξ − r is
infinitesimal. This settles the existence of sh(ξ). To prove its uniqueness, let’s
suppose that there exists r ′ ∈ R such as ξ ∼ r ∼ r ′ and r = r ′ . Since r and r ′ are
real numbers, this implies r ∼ r ′ , but this contradicts our hypothesis.
From this Theorem, we deduce immediately that every finite number can be
written as sum of one real and one infinitesimal number in a unique way.
Corollary 1.2.5. Any finite number ξ ∈ F has a unique representation in the following
form:
ξ = sh(ξ) + ɛ
where sh(ξ) ∈ R and ɛ = ξ − sh(ξ) is infinitesimal.
The notion of shadow can be extended to infinite numbers by setting:
• sh(ξ) = +∞ if ξ is infinite and positive;
• sh(ξ) = −∞ if ξ is infinite and negative.
Accordingly, we will adopt the usual conventional algebra of the extended real
line R ∪ {−∞, +∞}. As a result, we obtain that shadows satisfy compatibility
properties with respect to sums, products and quotients.
Proposition 1.2.6. For all numbers ξ, ζ, the following equalities hold:
1. sh(ξ + ζ) = sh(ξ) + sh(ζ);
2. sh(ξ · ζ) = sh(ξ) · sh(ζ);
3. sh(1/ξ) = 1/sh(ξ) whenever ξ is not infinitesimal.
The only exceptions are the following indeterminate forms:
(+∞) + (−∞); ∞ · 0; ∞
∞ ; 0
0
5

CHAPTER 1. Ω-CALCULUS
Proof. To prove part 1, if both numbers are finite, Corollary 1.2.5 ensures that we
can write
ξ + ζ = (sh(ξ) + ɛξ) + (sh(ζ) + ɛζ) = (sh(ξ) + sh(ζ)) + (ɛξ + ɛζ)
where both ɛξ and ɛζ are infinitesimal. Since sh(ξ) and sh(ζ) are real numbers and,
by Proposition 1.1.4, ɛξ + ɛζ is infinitesimal, by Corollary 1.2.5 we can conclude
sh(ξ + ζ) = sh(ξ) + sh(ζ)
If ξ is infinite and ζ is finite or if both ξ and ζ are infinite and have the same sign,
then the proposition is trivial.
To prove part 2 and 3, we can proceed in a similar way.
Similarly as with “classic” limits, in the indeterminate forms above, the resulting
shadows could be any element l ∈ R ∪ {−∞, +∞}, with the only restriction of sign
compatibility. In fact, for every given l, one can easily find ξi, ζi ∈ F such that
• sh(ξ1) = +∞, sh(ζ1) = −∞ and sh(ξ1 + ζ1) = l;
• sh(ξ2) = ∞, sh(ζ2) = 0 and sh(ξ2 · ζ2) = l;
• sh(ξ3) = ∞, sh(ζ3) = ∞ and sh( ξ3 ) = l; ζ3
• sh(ξ4) = 0, sh(ζ4) = 0 and sh( ξ4 ) = l. ζ4
If we extend the order relation to R ∪ {−∞, +∞} in the obvious way, by setting
−∞ ≤ l ≤ +∞ for all l ∈ R ∪ {−∞, +∞}, then shadows also satisfy a compatibility
property with respect to the order. More precisely,
Proposition 1.2.7. For all numbers ξ, ζ, sh(ξ) < sh(ζ) implies ξ < ζ.
The easy proof of this Proposition is left as exercise for the interested reader.
Remark 1.2.8. The above implication cannot be reversed: if ξ is finite and ɛ > 0 is
an infinitesimal number, ξ < ξ+ɛ, but sh(ξ) = sh(ξ+ɛ). On the other hand, the weak
inequality is preserved, and one can prove that ξ ≤ ζ if and only if sh(ξ) ≤ sh(ζ).
Up to now, we only considered the equivalence relation ∼ given by the notion
of being “infinitely close” (see Definition 1.2.3). Similarly, we can also consider the
relation of “finite closeness”:
ξ ∼f ζ if and only if ξ − ζ is finite.
It is readily seen that also ∼f is an equivalence relation. In the literature, the
equivalence classes relative to the two relations of closeness ∼ and ∼f, are called
monads and galaxies, respectively.
Definition 1.2.9. The monad of a number ξ is the set of all numbers that are
infinitely close to it:
mon(ξ) = {ζ ∈ F : ξ ∼ ζ}
6
The galaxy of a number ξ is the set of all numbers that are finitely close to it:
gal(ξ) = {ζ ∈ F : ξ ∼f ζ}

1.3. THE AXIOMS OF Ω-CALCULUS
In other words, mon(0) is the set of all infinitesimal numbers in F and gal(0) is
the set of all finite numbers.
In the following proposition, we itemize two of the basic properties of monads
and galaxies.
Proposition 1.2.10. 1. Any two monads (or galaxies) are either equal or disjoint;
2. The monad (or the galaxy) of a point ξ is the coset of ξ modulo mon(0) (modulo
gal(0), respectively). In other words, mon(ξ) = {ξ + ɛ : ɛ ∈ mon(0)} and
gal(ξ) = {ξ + ζ : ζ ∈ gal(0)}.
Proof. Part 1 is true because monads and galaxies are equivalence classes, while part
2 follows from the definition.
1.3 The Axioms of Ω-Calculus
Work out what the most beautiful
rules are.
Terry Pratchett
Now that all the necessary concepts are formally defined, we can introduce the
theory of Ω-Calculus. We choose to follow an axiomatic way: we start by giving
all the axioms of Ω-Calculus and then, in the following sections, we will study
thoroughly their properties and their consequences.
Our axioms begin with the fundamental request of the existence of a superreal
field R ∗ , whose existence is deeply connected to a new mathematical operation, that
we will call “Ω-limit”. R ∗ will be the natural setting for the study of calculus with
infinite and infinitesimal numbers.
Axiom 1 (Existence Axiom). Given a set Ω ⊃ R, there exists a (non-Archimedean)
field
R ∗ ⊃ R
such that every function ϕ : Pfin(Ω) → R has a unique “ Ω-limit” in R ∗ . This limit
is denoted by
lim
λ↑Ω ϕ(λ)
Moreover, we assume that every ξ ∈ R ∗ is the Ω-limit of some real function ϕ :
Pfin(Ω) → R.
The word “limit” bears a big cultural and emotive baggage, having been used
in mathematics for nearly two centuries as a fertile instrument in various fields of
research. We have chosen to use the same word for the concept of Ω-limit because
it encompasses the idea of evaluating the expression ϕ(λ) as the variable becomes
“bigger and bigger” in Ω. The readers should be aware that this limit bears few
similarities with the well known “ε-δ” limit used in calculus, and that even the basic
7

CHAPTER 1. Ω-CALCULUS
properties of Ω-limit must be deduced by the axioms of Ω-Calculus before they can
be used. Sometimes, those properties will be in open contrast with the ones of
classical limit, but this should not upset or discourage anyone: the two concepts are
fairly different, despite the similarity of their name.
Now, we proceed with the axioms of Ω-Calculus by asking that Ω-limit satisfies
some basic properties: the Ω-limit of eventually constant functions must assume the
expected value, and the field operations of R and R ∗ must be compatible.
Axiom 2 (Real Numbers Axiom). If ϕ is eventually constant, namely ∃λ0 ∈ Pfin(Ω)
such as ϕ(λ) = r for all λ ⊇ λ0, then
lim ϕ(λ) = r
λ↑Ω
Axiom 3 (Sum and Product Axiom). For all ϕ, ψ : Pfin(Ω) → R:
limλ↑Ω ϕ(λ) + limλ↑Ω ψ(λ) = limλ↑Ω(ϕ(λ) + ψ(λ))
limλ↑Ω ϕ(λ) · limλ↑Ω ψ(λ) = limλ↑Ω(ϕ(λ) · ψ(λ))
In the next definitions, we want to introduce a concept of “measure” of a set
E ⊆ R, that we will call the numerosity of E. The idea that lies behind numerosity
is that if E is finite, then the numerosity of E is the number of elements of E.
Then, if E is infinite, we will define its numerosity by extending this idea using the
Ω-limit. This new concept of “measure” will give rise to interesting results, that will
be thoroughly studied in Section 1.6.
Before we define numerosity, we find useful to introduce the following convention:
Definition 1.3.1. If E ⊆ Ω and ϕ : Pfin(Ω) → R, then we set
lim ϕ(λ) = lim ϕ(λ ∩ E)
λ↑E λ↑Ω
This definition allows us to simplify the notation of Ω-limit in a number of
situations, the first of which is the definition of numerosity.
Definition 1.3.2. The numerosity n(E) of a set E ⊆ Ω is defined by
n(E) = lim
λ↑E |λ|
We can think of numerosity as a way to “extend” the concept of “number of
elements” to infinite sets. In Section 1.6, we will give a precise meaning to this
intuitive statement. Now, we want to ensure that numerosity has at least one familiar
property, and the only way to do so is by introducing a new axiom. We will
distinguish this axiom from the axioms regarding Ω-limit by calling it by the name
of “numerosity axiom”.
Numerosity Axiom 1. If a, b ∈ Q, then
n([a, b))
n([0, 1))
= (b − a)
This axiom rules that, for intervals with rational extrems, once we “normalize”
their numerosity by a meaningful constant factor, the result is the length of the
interval. We signal that we couldn’t have ruled a more general axiom that imposes
the equality for arbitrary intervals, but the proof of this statement requires the
introduction of new concepts depending upon Ω-limit. The interested reader could
easily deduce it as a consequence of Proposition 6.4.5.
8

1.4 First properties of Ω-limit
1.4. FIRST PROPERTIES OF Ω-LIMIT
In the previous section, we have introduced the axioms of Ω-Calculus: most of
them regard a new mathematical operation, called Ω-limit, which is quite different
from the limit used in classical calculus. In this section, we will begin to study the
behaviour of this Ω-limit, highlighting some of its main properties.
The first property we will show is that, if two functions are different for all
λ ∈ Pfin(Ω), then their Ω-limits will be different, too. This is quite in contrast with
what happens to usual limits of functions.
Proposition 1.4.1. If ϕ(λ) = ψ(λ) for all λ ∈ Ω, then limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ).
Proof. Begin by noticing that, by hypothesis,
for all λ ∈ Pfin(Ω), so that the function
ϕ(λ) − ψ(λ) = 0
η(λ) =
1
ϕ(λ) − ψ(λ)
is well defined for all λ ∈ Pfin(Ω). Moreover, the equality
η(λ)(ϕ(λ) − ψ(λ)) = 1
holds for all λ ∈ Pfin(Ω). Axiom 2 and 3 ensure that
lim η(λ) · lim(ϕ(λ)
− ψ(λ)) = lim η(λ)(ϕ(λ) − ψ(λ)) = 1
λ↑Ω λ↑Ω λ↑Ω
and this implies, in particular, that
so the thesis follows.
lim(ϕ(λ)
− ψ(λ)) = 0
λ↑Ω
The second property is that if a function assumes only a finite number of values,
then its Ω-limit must be one of those values.
Proposition 1.4.2. If E = {x1, . . . , xn} ⊂ R is finite and ϕ(λ) ∈ E for all λ ∈
Pfin(Ω), then limλ↑Ω ϕ(λ) ∈ E.
Proof. By hypothesis,
lim
λ↑Ω
n
(ϕ(λ) − xi) = 0
i=1
Explicitating the formula, we have that exists i such as limλ↑Ω ϕ(λ) = xi, as we
wanted.
For further development of the theory, for every couple of functions ϕ and ψ :
Pfin(Ω) → R, we find useful to define a set Qϕ,ψ that contains all λ ∈ Pfin(Ω) such
as ϕ(λ) = ψ(λ). These sets will play a major role in understanding the behaviour
of Ω-limit.
9

CHAPTER 1. Ω-CALCULUS
Definition 1.4.3. For all ϕ, ψ : Pfin(Ω) → R, we define
Clearly, Qϕ,ψ ⊆ Pfin(Ω).
Qϕ,ψ = {λ ∈ Pfin(Ω) : ϕ(λ) = ψ(λ)}
In the next proposition, we will explicitely show the relation between the sets
Qϕ,ψ and Ω-limit.
Proposition 1.4.4. For all ϕ, ψ : Pfin(Ω) → R,
1. if limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ), then Qϕ,ψ = ∅;
2. If limλ↑Ω ϕ(λ) = 0 and Qϕ,c0 ⊆ Qψ,c0 (e.g. ψ vanishes where ϕ does), then
limλ↑Ω ψ(λ) = 0;
3. If limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ), then for all η, γ : Pfin(Ω) → R, if Qϕ,ψ ⊆ Qη,γ,
then limλ↑Ω η = limλ↑Ω γ.
Proof. Part 1 of this Proposition is the contrapositive of Proposition 1.4.1.
To prove part 2, define
ϕ(λ) if ϕ(λ) = 0
δ(λ) =
1 if ϕ(λ) = 0
Since δ(λ) = 0 for all λ ∈ Pfin(Ω), limλ↑Ω δ(λ) = ξ = 0. It’s easy to verify that
for all λ ∈ Pfin(Ω). This implies
(δ(λ) − ϕ(λ))ψ(λ) = 0
0 = lim
λ↑Ω (δ(λ) − ϕ(λ))ψ(λ) = lim
λ↑Ω (δ(λ) − ϕ(λ)) · lim
λ↑Ω ψ(λ) = ξ · lim
λ↑Ω ψ(λ)
and, since ξ = 0, then limλ↑Ω ψ(λ) = 0.
To prove part 3, define the function ϕ ′ (λ) = ϕ(λ) − ψ(λ). By hypothesis,
limλ↑Ω ϕ ′ (λ) = 0, and it’s clear that Qϕ ′ ,c0 = Qϕ,ψ. Now, define η ′ (λ) = η(λ) − γ(λ).
It’s easy to verify that Qϕ ′ ,η ′ = Qϕ,ψ and, using part 2 of this proposition, we conclude
that limλ↑Ω η ′ (λ) = 0 or, in other words,
The thesis follows.
lim η
λ↑Ω ′ (λ) = lim(η(λ)
− γ(λ)) = lim η(λ) − lim γ(λ) = 0
λ↑Ω λ↑Ω λ↑Ω
Remark 1.4.5. We want to spend a few words on the content of Proposition 1.4.4.
For any two functions ϕ and ψ, we have defined the set Qϕ,ψ and we have shown
that this set plays a fundamental role in determining the Ω-limit of these functions:
if we know, for instance, that ϕ and ψ have the same Ω-limit, then every other pair
of functions η and γ who coincide at least on every λ ∈ Qϕ,ψ satisfy
10
lim η(λ) = lim γ(λ)
λ↑Ω λ↑Ω

1.4. FIRST PROPERTIES OF Ω-LIMIT
A consequence of this result is that, if ϕ and ψ have the same Ω-limit, we can
change the values of ϕ(λ) even for all λ ∈ Qϕ,ψ, but the function obtained will have
the same Ω-limit as ϕ.
We can go even further: if a function ϕ is well defined only for all λ in a qualified
set Q, this is sufficient to ensure that ϕ has an Ω-limit. In fact, we can define
arbitrarily the values of ϕ(λ) for all λ ∈ Q c , and the corresponding functions will
always have the same Ω-limit. This feature presents a strong analogy to the case of
real, convergent sequences {xn}n∈N: even if we change a finite number of the xn, the
limit of the sequence doesn’t change and, if there is m ∈ N such that the sequence is
defined only for n ≥ m, then this is sufficient to ensure that the limit of the sequence
is well defined.
Since the sets Qϕ,ψ satisfying the hypothesis that ϕ and ψ have the same Ω-limit
play such an important role in the theory, we will give them a name.
Definition 1.4.6. We will call Q ⊆ P(Pfin(Ω)) the set of all Qϕ,ψ satisfying
limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ):
Q =
Qϕ,ψ : lim
λ↑Ω ϕ(λ) = lim
λ↑Ω ψ(λ)
We will say that a set Q ∈ Q is qualified, and we will call Q the set of qualified
sets.
All the sets Q ∈ Q are, so to speak, the sets that “count” for the Ω-limit or, in
other words, the Ω-limit of a functon depends only on its behaviour on the elements
of a set Q ∈ Q. This idea is precised in the following Corollary.
Corollary 1.4.7. For all ϕ, ψ : Pfin(Ω) → R, limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ) if and only
if Qϕ,ψ ∈ Q.
Proof. One of the implications is true by definition, so we will focus on the other.
If Qϕ,ψ ∈ Q, then there are two functions η and γ such as limλ↑Ω η = limλ↑Ω γ and
Qη,γ = Qϕ,ψ. Now we can apply part 3 of Proposition 1.4.4 to obtain limλ↑Ω ϕ(λ) =
limλ↑Ω ψ(λ).
The family of sets Q has some additional properties with respect to set operations:
they are summarized in the next Proposition.
Proposition 1.4.8. Q satisfies the following properties:
1. If A ∈ Q and B ⊇ A, then B ∈ Q;
2. If A, B ∈ Q, then A ∩ B ∈ Q;
3. If A ∪ B ∈ Q, then A ∈ Q or B ∈ Q;
4. A ∈ Q ⇔ A c ∈ Q;
5. If F is finite (e.g. F = {λ1, . . . , λn}), then F ∈ Q;
11

CHAPTER 1. Ω-CALCULUS
6. For all r ∈ R, Fr = {λ ∈ Pfin(Ω) : r ∈ λ} ∈ Q;
7. ∅ ∈ Q and Pfin(Ω) ∈ Q.
Proof. The first part of this Proposition follows from Proposition 1.4.4.
To prove part 2, we will show that, if A = Qϕ,ψ and B = Qη,γ, then A∩B = Qδ,φ,
and δ and φ have the same Ω-limit. If A ⊆ B or B ⊆ A, the thesis follows from
part 1. Let’s suppose that this is not the case. Let’s call ξ = limλ↑Ω ϕ(λ) and
ζ = limλ↑Ω η(λ). If ξ = ζ, Axiom 1 ensures that there is a function ν such as
limλ↑Ω ν(λ) = ξ −ζ. If we define η ′ = η +ν and γ ′ = γ +ν, we have that Qη ′ ,γ ′ = Qη,γ
and
lim η
λ↑Ω ′ = lim γ
λ↑Ω ′ = ζ + ξ − ζ = ξ
We have shown that, without loss of generality, we can suppose
Now, it’s clear that
and
lim ϕ(λ) = lim ψ(λ) = lim η(λ) = lim γ(λ)
λ↑Ω λ↑Ω λ↑Ω λ↑Ω
Qϕ−η,ψ−γ = Qϕ,ψ ∩ Qη,γ
lim(ϕ(λ)
− η(λ)) = lim(ψ(λ)
− γ(λ)) = 0
λ↑Ω λ↑Ω
and that’s what we wanted to prove.
The proof of part 3 is by converse. Let’s suppose A ∈ Q and B ∈ Q. From
the definition we have that, for all functions ϕ and ψ, the hypothesis Qϕ,ψ = A or
Qϕ,ψ = B implies limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ). Since the implication holds if both the
premises are true, we have that, if for ϕ and ψ the equality Qϕ,ψ = A ∪ B holds,
then limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ), and so we conclude A ∪ B ∈ Q, as we wanted.
To prove the first implication of part 4, let’s suppose A ∈ Q, so that there exist
ϕ, ψ such as A = Qϕ,ψ and limλ↑Ω ϕ = limλ↑Ω ψ. Let’s define the functions:
and
η(λ) =
γ(λ) =
ϕ(λ) if λ ∈ A
1 if λ ∈ A c
ϕ + 1 if λ ∈ A
1 if λ ∈ A c
Clearly, Qη,γ = A c and, by part 3 of Proposition 1.4.4, limλ↑Ω η(λ) = limλ↑Ω ϕ(λ),
while limλ↑Ω γ(λ) = limλ↑Ω ϕ(λ)+1. This shows that A c ∈ Q. Conversely, if A c ∈ Q,
we can apply the same argument to show that A = (A c ) c ∈ Q.
To prove part 5, we can consider the function c0 and the function
c
0 if λ ∈ F
ϕ(λ) =
1 if λ ∈ F
By definition, we have Qc0,ϕ = F c , so ϕ is eventually constant (take λ0 = n
i=1 λi)
and, by Axiom 2, limλ↑Ω ϕ(λ) = 0. This implies F c ∈ Q and, by part 4, F ∈ Q.
12

1.4. FIRST PROPERTIES OF Ω-LIMIT
To prove part 6, we have that, if we have two functions ϕ and ψ such that Qϕ,ψ =
Fr, then ϕ(λ)−ψ(λ) = 0 is eventually true (take λ0 = {r}), so Axiom 2 ensures that
limλ↑Ω(ϕ(λ)−ψ(λ)) = 0, and Axiom 3 allows to conclude limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ).
An application of Corollary 1.4.7 gives us the thesis.
Part 7 follows from Axiom 2 and from part 4 of this Proposition.
A family of subsets satisfying the seven properties itemized in this proposition
is called a nonprincipal fine ultrafilter on Pfin(Ω). This important notion is crucial
in constructing models for Ω-Calculus, and will be further studied in Sections 1.10
and 1.11.
As final result of this section, we will show that R ∗ has a natural structure of
ordered field whose order extends the order on R.
Proposition 1.4.9. R ∗ is an ordered field whose positive part (R ∗ ) + is
(R ∗ ) + =
lim ϕ(λ) : ϕ(λ) > 0 ∀λ ∈ Pfin(Ω)
λ↑Ω
Proof. From Axiom 3, we have that (R ∗ ) + is closed under sum and products. More-
over, if we set
we have
(R ∗ ) − =
(R ∗ ) − =
lim ϕ(λ) : −ϕ(λ) > 0 ∀λ ∈ Pfin(Ω)
λ↑Ω
lim ϕ(λ) : ϕ(λ) < 0 ∀λ ∈ Pfin(Ω)
λ↑Ω
We need only to show that (R∗ ) − , {0} and (R∗ ) + form a partition of R∗ . First
of all, notice that Proposition 1.4.1 ensures that those sets are pairwise disjoint.
Now, for every ϕ : Pfin(Ω) → R, define
ϕ +
ϕ(λ) if ϕ(λ) > 0
(λ) =
1 if ϕ(λ) ≤ 0
and
It’s easy to see that
ϕ − (λ) =
ϕ(λ) if ϕ(λ) < 0
−1 if ϕ(λ) ≥ 0
ϕ(λ)(ϕ(λ) − ϕ + (λ))(ϕ(λ) − ϕ − (λ)) = 0
for all λ ∈ Pfin(Ω), so that limλ↑Ω ϕ(λ) = 0 implies
or
as we wanted.
lim ϕ(λ) = lim ϕ
λ↑Ω λ↑Ω + (λ) ∈ (R ∗ ) +
lim ϕ(λ) = lim ϕ
λ↑Ω λ↑Ω − (λ) ∈ (R ∗ ) −
13

CHAPTER 1. Ω-CALCULUS
1.5 The Transfer Principle
In this section, we want to formulate a transfer principle for the notion of Ω-limit.
Since a precise formulation of the transfer principle is possible only after the introduction
of some logic formalism, we settle for an informal version where the
important concept of of “elementary property” is left undefined.
Transfer Principle (Informal version). An “elementary property” P is satisfied by
real numbers ϕ1(λ), . . . , ϕk(λ) for all λ in a qualified set if and only if P is satisfied
by the Ω-limits limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕk(λ).
In other words, Transfer Principle tells us that for all “elementary properties”
P , the set QP = {λ ∈ Pfin(Ω) : P (ϕ1(λ), . . . , ϕk(λ)) holds} ∈ Q if and only if
P (limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕk(λ)) holds. If P is a property such as QP ∈ Q, we will
say that P holds in a qualified set or that P holds almost everywere. Sometimes,
we will also use the abbreviation “P holds a.e.”
For now, the Transfer Principle can be taken only at an an informal, intuitive
level, because we have no formal definition of an “elementary property” (and this
is the reason why we always put “elementary property” between quotation marks).
The content of Transfer Principle is made precise only when a rigorous definition of
“elementary property” is given, and this will be done in Chapter 5 of this paper.
Now we want to prove the Transfer Principle for some fundamental “elementary
properties” we will use quite often.
Proposition 1.5.1 (Transfer of inequalities). Let ϕ, ψ : Pfin(Ω) → R. Then
1. ϕ(λ) = ψ(λ) a.e. if and only if limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ);
2. ϕ(λ) < ψ(λ) a.e. if and only if limλ↑Ω ϕ(λ) < limλ↑Ω ψ(λ);
3. ϕ(λ) ≤ ψ(λ) a.e. if and only if limλ↑Ω ϕ(λ) ≤ limλ↑Ω ψ(λ).
Proof. To prove part 1, first notice that that, in both cases, Q = {λ ∈ Pfin(Ω) :
ϕ(λ) = ψ(λ)} ∈ Q, and Proposition 1.4.8 ensures that Qϕ,ψ = Q c ∈ Q. Now, the
thesis follows from Corollary 1.4.7.
To prove part 2, suppose that ϕ(λ) < ψ(λ) a.e. We can define
δ(λ) = ψ(λ) − ϕ(λ)
and notice that, by hypothesis, Q = {λ ∈ Pfin(Ω) : δ(λ) > 0} ∈ Q. If limλ↑Ω δ(λ) ≤
0, then we would have Q c ∈ Q, but once again this contradicts Proposition 1.4.8.
The proof of part 3 can be obtained in a similar way.
The reader should not think that all the properties of real numbers are elementary.
Indeed, we can even show explicit counterexamples of properties that are not
elementary.
Example 1.5.2. As a property of real numbers, the property P (x) saying “x is a
natural number” is not elementary, since |λ| ∈ N for all λ ∈ Pfin(Ω), but clearly
14

1.6. NUMEROSITY
limλ↑Ω |λ| ∈ N. More generally, we will soon see in Proposition 1.7.6 that the property
P (x) saying “x ∈ E” is elementary if and only if E is finite. We remark that,
in this case, the property x ∈ E is equivalent to “given the real numbers x1, . . . , xn,
then x = x1 or x = x2 . . . or x = xn”.
We signal that the property ϕ(λ) ∈ N is elementary if considered as a property
of numbers and sets. In fact, in the context of a theory that rules the Ω-limit of
sets, and not only of real numbers, it can be proved that the property ϕ(λ) ∈ N,
considered as a property of numbers and sets, transfers to limλ↑Ω ϕ(λ) ∈ limλ↑Ω cN(λ).
Of course, this statement has no meaning until we define within the theory what is
the Ω-limit of a function ϕ : Pfin(Ω) → P(R). Even if this will be done in Section
1.7, we remand the study of a more general version of Transfer Principle to Section
4.4 of this paper, where we will have a more general theory of Ω-limit and we will
be able to prove the validity of Transfer of membership for arbitrary functions.
1.6 Numerosity
Numerosity is a new way to “count” the number of elements of a set by using Ωlimit.
In this section, we will show some of its properties and highlight the most
important of them.
First of all, we will show that, if F ⊂ R is a finite set, then n(F ) = |F |. This idea
has already been expressed informally in Section 1.3, while introducing the definition
of numerosity.
Proposition 1.6.1. If F ⊂ R is a finite set, then n(F ) = |F |.
Proof. Since F is finite, F ∈ Pfin(Ω). This ensures that, for all λ ⊇ F , F ∩ λ = F .
By definition of numerosity, we obtain that the equality |F ∩ λ| = |F | is eventually
true, so the thesis follows.
Proposition 1.6.1 ensures that, when F is finite, then numerosity satisfies the
basic intuitive properties about counting: finite additivity and monotony. We already
know that, when dealing with infinite sets, cardinality loses the property of
monotony: for example, N ⊂ Z but |N| = |Z|. It’s only natural to ask whether this
happens also for numerosity. It turns out that, thanks to the Ω-limit and Transfer
Principle, numerosity is always finitely additive and monotone.
Proposition 1.6.2. If E1, E2 ⊆ R are two disjoint sets, then n(E1 ∪ E2) = n(E1) +
n(E2).
Proof. By hypothesis,
|λ ∩ (E1 ∪ E2)| = |λ ∩ E1| + |λ ∩ E2|
holds for all λ ∈ Pfin(Ω), so we can apply Transfer Principle and conclude.
This result can be easily generalized to the case of finite unions of disjoint sets.
15

CHAPTER 1. Ω-CALCULUS
Corollary 1.6.3 (Finite additivity of numerosity). For all finite sequences of real
subsets {Ei}i∈{1,...,n} satisfying Ei ∩ Ej = ∅ whenever i = j,
n
n
n = n(Ei)
i=1
Ei
In the next Proposition, we will prove that numerosity is monotone.
Proposition 1.6.4 (Monotony of numerosity). For all E, F ⊆ R, if E ⊂ F , then
n(E) < n(F ).
Proof. We can write F = E ∪ (F \ E) and, by hypothesis, F \ E = ∅. By Corollary
1.6.3, we have that n(F ) = n(E) + n(F \ E). If F \ E is a finite set, we already
know n(F \ E) > 0; if F \ E is infinite, then pick x ∈ F \ E: clearly, n(F \ E) ≥
|(F \ E) ∩ {x}| > 0. This shows that in both cases n(F \ E) > 0, so the proof is
concluded.
As a consequence of monotony, we deduce that infinite sets have infinite numerosity.
Corollary 1.6.5. If E is infinite, then n(E) is an infinite number.
Proof. Thanks to Proposition 1.6.4, n(E) > n(F ) for all finite F ⊂ E and, by the
fact that E contains finite subsets with arbitrary number of elements, this means
n(E) > n for all n ∈ N.
We want to remark that the facts that numerosity is always finitely additive
and monotone can be seen as consequences of the Transfer Principle for equalities
and inequalities: thanks to the Ω-limit, the properties of the finite cardinality are
extended to numerosity of arbitrary sets.
The monotony of numerosity seems to be close to Euclid’s idea that the whole is
greater than the parts, an idea that has been somehow challenged by the counting
methods of cardinality and of ordinals. In this regard, we find appropriate the
following quotation:
The possibility that whole and part may have the same number of terms is, it must
be confessed, shocking to common sense. (Russell, 1903, Principles of Mathematics,
p. 358)
We conclude this discussion with a well known example:
Example 1.6.6. The set of natural numbers is the disjoint union of the set E
of even numbers and the set O of odd numbers. By Proposition 1.6.2, we know
that n(N) = n(E) + n(O), but our theory doesn’t tell if, for instance n(E) = n(O).
This difficulty is due to the fact that, in the axiom of numerosity, we focused on
a property that links the numerosity to the metric of R. If we were interested in
other applications of numerosity, we could have chosen different axioms to ensure
the validity of some other properties useful for our purposes.
We signal that, once we have shown a model for Ω-Calculus, we will be able to
calculate the exact numbers n(E) and n(O).
16
i=1

1.6. NUMEROSITY
In the previous part of this Section, we have shown some properties of numerosity
that are independent from Numerosity Axiom 1. Now, we want to focus on the
behaviour of numerosity in the case of real subsets. At first, thanks to Proposition
1.6.4, we will show how the equation of Numerosity Axiom 1 can be extended to all
a, b ∈ R.
Proposition 1.6.7. For all a, b ∈ R, then
n([a, b))
n([0, 1))
∼ b − a
Proof. If a and b ∈ Q, then there’s nothing to prove. Otherwise, for all n ∈ N we
can find a + n and b + n ∈ Q such that a + n < a and b + n > b and
0 ≤ (b + n − a + n ) − (b − a) < 1
n
(1.1)
On the other hand, we can also find a− n and b− n ∈ Q such that a− n < a and b− n > b
and
0 ≤ (b − a) − (b − n − a − n ) < 1
(1.2)
n
Putting together equations 1.1 and 1.2, we obtain
By Proposition 1.6.4, we deduce
lim(b
λ↑Ω +
|λ| − a+
|λ| ) ∼ lim
λ↑Ω (b− |λ| − a−
|λ| ) ∼ b − a (1.3)
n([a − n , b − n ))
n([0, 1))
n([a, b))
<
n([0, 1)) < n([a−n , b− n ))
n([0, 1))
for all n ∈ N and, by equation 1.3, we can conclude
n([a, b))
n([0, 1))
∼ b − a
This proposition, while helpful to determine the numerosity of arbitrary intervals,
has also an interesting consequence:
Corollary 1.6.8. Numerosity is not invariant with respect to translations.
This consideration inspires the following question: are there some transformations
of the real line that preserve numerosity? If the answer is yes, then how can
they be characterized? In other words, besides the identity, what are the “isometries”
of R with respect to numerosity?
Indeed, we can find some characterizations of such functions, but they are rather
nondescriptive and don’t tell us wether there are some functions that satisfy them.
One of such characterizations is presented in the following Proposition.
17

CHAPTER 1. Ω-CALCULUS
Proposition 1.6.9. Let f : R → R satisfy |f(F )| = |F | for all finite subsets of R.
If for all E ⊆ R the set {λ ∈ Pfin(Ω) : |f(E ∩ λ)| = |f(E) ∩ λ|} is qualified, then n
is invariant with respect to f.
Proof. By hypothesis, |E ∩ λ| = |f(E ∩ λ)| and, since |f(E ∩ λ)| = |f(E) ∩ λ| almost
everywhere, by transfer of equality we conclude
Hence the thesis.
lim |f(E) ∩ λ| = lim |f(E ∩ λ)| = lim |E ∩ λ|
λ↑Ω λ↑Ω λ↑Ω
It’s clear that the functions characterized by Proposition 1.6.9 depend on the
ultrafilter Q, hence the difficulty in determining their existence.
Numerosity presents some other interesting challenges and open questions, but
we will remand them to Section 3.4 of this paper, when we will have at our disposal
more mathematical tools.
1.7 Extensions of sets
The Ω-limit is a mathematical instrument that allows the construction of hyperreal
numbers out of real numbers. In the same spirit, a notion of Ω-limit can also be
given for functions ϕ : Pfin(Ω) → P(R): as a result, we will be able to “extend” the
subsets of R into subsets of R ∗ by using the Ω-limit.
We start by defining what we intend as the Ω-limit of a sequence of sets.
Definition 1.7.1. Let {Eλ}λ∈Pfin(Ω) be a family of nonempty subsets of R. We
define
lim Eλ =
λ↑Ω
lim ϕ(λ) : ϕ(λ) ∈ Eλ
λ↑Ω
and
lim c∅(λ) = ∅
λ↑Ω
In the same spirit, if E is a subset of Ω, we define
lim Eλ = lim ϕ(λ ∩ E) : ϕ(λ ∩ E) ∈ Eλ
λ↑E λ↑Ω
Thanks to Transfer Principle, the definition of limλ↑Ω Eλ can be given by an
“almost everywhere” formulation. More precisely, we will show that
Proposition 1.7.2. If {Eλ}λ∈Pfin(Ω) is a family of nonempty subsets of R, then
18
lim Eλ = lim ϕ(λ) : ϕ(λ) ∈ Eλ a.e.
λ↑Ω λ↑Ω

1.7. EXTENSIONS OF SETS
Proof. By hypothesis, for all λ ∈ Pfin(Ω) we can pick eλ ∈ Eλ. Now, suppose that
ϕ(λ) ∈ Eλ a.e.: then, if we define a function
ϕ ′
ϕ(λ) if ϕ(λ) ∈ Eλ
(λ) =
eλ if ϕ(λ) ∈ Eλ
we have that ϕ ′ (λ) = ϕ(λ) almost everywhere and, by Transfer of equality, we conclude
limλ↑Ω ϕ ′ (λ) = limλ↑Ω ϕ(λ). Now observe that, by construction, limλ↑Ω ϕ ′ (λ) ∈
limλ↑Ω Eλ, so the thesis follows.
Thanks to Definition 1.7.1, we can now define two kinds of extensions of sets:
the first, more generic, is the natural extension.
Definition 1.7.3. The natural extension of a set E ⊆ R is given by
E ∗
= lim cE(λ) = lim ϕ(λ) : ϕ(λ) ∈ E
λ↑Ω λ↑Ω
where cE(λ) is the function identically equal to E.
Then, we can define another extension, more restrictive than the one we’ve already
seen, but with more interesting properties: the hyperfinite extension.
Definition 1.7.4. The hyperfinite extension of a set E ⊆ R is given by
E ◦ = lim
λ↑E λ
It’s worthy to explicitate this definition:
E ◦
= lim ϕ(λ ∩ E) : ϕ(λ ∩ E) ∈ λ ∩ E
λ↑Ω
From the two definitions above, it’s clear that E ⊆ E ◦ ⊆ E ∗ .
Now, we want to study some elementar properties of those extensions of sets,
and their behaviour under some basic set operations. The first property we want to
show is that equality is preserved under natural and hyperfinite extension.
Proposition 1.7.5. Let E, F ⊆ R. E = F if and only if E ∗ = F ∗ if and only if
E ◦ = F ◦ .
Proof. If E = F , then it’s trivial that E ∗ = F ∗ and E ◦ = F ◦ .
Now, we want to show that, if E = F , then E ∗ = F ∗ and E ◦ = F ◦ . Let’s suppose
that E ⊆ F , so that we can choose x satisfying x ∈ E and x ∈ F . In particular, we
have that, if ϕ(λ) ∈ F for all λ ∈ Pfin(Ω), then ϕ(λ) = x and so x ∈ F ◦ and x ∈ F ∗ ,
so that E ◦ ⊆ F ◦ and E ∗ ⊆ F ∗ . This allows to conclude E ◦ = F ◦ , as desired.
In the next two Propositions, we will see that the only sets for which it holds the
equality chain E = E ◦ = E ∗ are finite sets. We will split the proof of this statement
in two parts: at first, we will prove that E = E ∗ if and only if E is finite, and then
that E ◦ = E ∗ if and only if E is finite.
19

CHAPTER 1. Ω-CALCULUS
Proposition 1.7.6. Let E ⊆ R. E = E ∗ if and only if E is finite.
Proof. If E is finite, we can apply Proposition 1.4.2 and obtain immediately E ∗ = E.
If E is infinite, then it’s not bounded or it’s bounded and have an accumulation
point p ∈ R. In the first case, we can find a function ϕ : Pfin(Ω) → E such as
for all n ∈ N one of the relations ϕ(λ) > n or −ϕ(λ) > n eventually holds, and so
limλ↑Ω ϕ(λ) is an infinite number that doesn’t belong to E.
In the second case, we can find a function ϕ : Pfin(Ω) → E such as |ϕ(λ) − p| <
1/|λ| and ϕ(λ) = p. Taking the Ω-limit of both sides and calling ξ = limλ↑Ω ϕ(λ),
we have that |ξ − p| < 1/k for all k ∈ N, so that |ξ − p| is infinitesimal or, in other
words, ξ ∈ mon(p). Now, ϕ(λ) = p for all λ ∈ Pfin(Ω), so ξ = p, and this implies
ξ ∈ R. Since ξ ∈ E ∗ and ξ ∈ E, we conclude E ∗ = E, as we wanted.
Proposition 1.7.7. Let E ⊆ R. E ◦ = E ∗ if and only if E is finite.
Proof. If E is finite, then, by Proposition 1.7.6, we have E = E ∗ ⊇ E ◦ ⊇ E, and
the thesis follows.
If E is infinite, we want to show that E ∗ \ E ◦ = ∅ or, in other words, we want to
show that there is a function ψ : Pfin(Ω) → E such as for every ϕ : Pfin(Ω) → E we
have ζ = limλ↑Ω ψ(λ) ∈ E ◦ . To do so, first notice that, for all λ ∈ Pfin(Ω), E \λ = ∅,
because E infinite and λ is finite. This implies that there exists a function ϕ such as
ϕ(λ) ∈ E \ (λ ∩ E) for all λ ∈ Pfin(Ω). Now, since every ξ ∈ E ◦ is the Ω-limit of a
function ϕ such as ϕ(λ ∩ E) ∈ λ ∩ E, it’s clear that, by definition, for all such ϕ we
have ψ(λ) = ϕ(λ ∩ E). This and Proposition 1.4.1 ensures that ζ = limλ↑Ω ψ(λ) = ξ
for all ξ ∈ E ◦ . By definition, we have also ζ ∈ E ∗ , so that ζ ∈ E ∗ \ E ◦ , as we
wanted.
It turns out that the relation between E ◦ and E ∗ can be further characterized:
Proposition 1.7.8. for all E ⊆ R, E ◦ = E ∗ ∩ R ◦ .
Proof. First, let’s suppose that ξ ∈ E ∗ ∩R ◦ . We can find a function ϕ : Pfin(Ω) → E
such that limλ↑Ω ϕ(λ) = ξ and a function ψ : Pfin(Ω) → R such that ψ(λ) ∈ λ and
limλ↑Ω ψ(λ) = ξ. Since ϕ and ψ have the same Ω-limit, we have that Qϕ,ψ ∈ Q and
so ψ(λ) ∈ E for all λ ∈ Qϕ,ψ, but this implies ψ(λ) ∈ E ∩ λ for those λ. Thanks to
Proposition 1.7.2, we can safely conclude that ξ ∈ E ◦ . This proves E ◦ ⊇ E ∗ ∩ R ◦ .
The other inclusion is straightforward: if ξ ∈ E ◦ , then by definition we have
ξ ∈ E ∗ and ξ ∈ R ◦ , so that ξ ∈ E ∗ ∩ R ◦ .
Thanks to this Proposition, a lot of results about natural extension of sets will
apply also to hyperfinite extension. Now, we will show that the operation of natural
extension commutes with set operations and, as a consequence, we will obtain that
the same result holds for hyperfinite extension.
Proposition 1.7.9. for all E, F ⊆ R, the following are true:
20
1. E ∗ ∩ F ∗ = (E ∩ F ) ∗ ;
2. E ⊆ F implies E ∗ ⊆ F ∗ ;

3. E ∗ ∪ F ∗ = (E ∪ F ) ∗ ;
4. (E c ) ∗ = (E ∗ ) c ;
5. E ∗ \ F ∗ = (E \ F ) ∗ .
Proof. Part 1 is straightforward, once we notice that
1.7. EXTENSIONS OF SETS
{ϕ : Pfin(Ω) → E ∩ F } = {ϕ : Pfin(Ω) → E} ∩ {ϕ : Pfin(Ω) → F }
To prove part 2, notice that, by part 1 and Proposition 1.7.6,
E ∗ ⊆ F ∗ ⇔ E ∗ = E ∗ ∩ F ∗ = (E ∩ F ) ∗ ⇔ E = E ∩ F ⇔ E ⊆ F
As for part 3, if E or F is empty, the thesis follows. Let’s suppose that both sets
are nonempty, and pick e ∈ E and f ∈ F . For every function ϕ : Pfin(Ω) → E ∪ F ,
define:
and
ϕ ′ (λ) =
ϕ ′′ (λ) =
ϕ(λ) if ϕ(λ) ∈ E
e if ϕ(λ) ∈ E
ϕ(λ) if ϕ(λ) ∈ F
f if ϕ(λ) ∈ F
Since (ϕ(λ) − ϕ ′ (λ)) · (ϕ(λ) − ϕ ′′ (λ)) = 0, we have that limλ↑Ω ϕ(λ) = limλ↑Ω ϕ ′ (λ)
or limλ↑Ω ϕ(λ) = limλ↑Ω ϕ ′′ (λ). In both cases, limλ↑Ω ϕ(λ) ∈ E ∗ ∪ F ∗ , and so (E ∪
F ) ∗ ⊆ E ∗ ∪ F ∗ . The reverse inclusion follows from the fact that, by part 2, both
E ∗ ⊆ (E ∪ F ) ∗ and F ∗ ⊆ (E ∪ F ) ∗ .
To prove part 4, we can use that (E ∗ ) c is the only set X such as E ∗ ∩ X = ∅
and E ∗ ∪ X = R ∗ . By using part 1 and 3 of this Proposition, it’s easily checked
that X = (E c ) ∗ satisfies both conditions: E ∗ ∩ (E c ) ∗ = (E ∩ E c ) ∗ = ∅ ∗ = ∅ and
E ∗ ∪ (E c ) ∗ = (E ∪ E c ) ∗ = R ∗ .
Corollary 1.7.10. for all E, F ⊆ R, the following are true:
1. E ◦ ∩ F ◦ = (E ∩ F ) ◦ ;
2. E ⊂ F implies E ◦ ⊆ F ◦ ;
3. E ◦ ∪ F ◦ = (E ∪ F ) ◦ ;
4. (E c ) ◦ = (E ◦ ) c ;
5. E ◦ \ F ◦ = (E \ F ) ◦ .
Proof. This is a consequence of Proposition 1.7.9 and Proposition 1.7.8.
Now, we want to show how natural extension behaves with respect to countable
union.
21

CHAPTER 1. Ω-CALCULUS
Proposition 1.7.11. Let {En}n∈N be disjoint subsets of R. Then,
En ⊆ lim
∗ n∈N
En ⊆
λ↑Ω
n≤|λ|
Moreover, the first inclusion is an equality if and only if
n∈N En is finite, and the
second inclusion is an equality if and only if En = ∅ is eventually true.
Proof. If
n∈N En is a finite set, then the thesis follows from Proposition 1.7.6 and,
if En = ∅ is eventually true, then the thesis is trivial.
Suppose that those are not the cases. Omitting redundant terms, we can suppose
that |λ1| < |λ2| implies
En ⊂
n≤|λ1|
n≤|λ2|
We can then define a function ϕ : Pfin(Ω) → R in such a way that
⎛
ϕ(λ) ∈ ⎝
En \
⎞
⎠
n≤|λ|+1
for all λ ∈ Pfin(Ω). By definition, limλ↑Ω ϕ(λ) ∈ En for all n ∈ N, and this concludes
the first part of the proof.
As for the second part, we can define a function ψ : Pfin(Ω) → R in such a way
that
⎛
ψ(λ) ∈ ⎝
En \
⎞
⎠
n∈N
n∈N
En
n≤|λ|
n≤|λ|
for all λ ∈ Pfin(Ω). For such a ψ it holds limλ↑Ω ψ(λ) ∈ limλ↑Ω
thesis follows.
En
En
En
n≤|λ| En, so the
Now, we want to focus on the properties of hyperfinite sets that are not shared by
the natural extension of sets. Maybe, the most important of them is the following:
Proposition 1.7.12. for all E ⊆ R, E ◦ has maximum and minimum.
Proof. The two functions max and min are well defined for all λ ∈ Pfin(Ω). If we
take their limit, by definition we have
and
η = lim
λ↑E max(λ) ∈ E ◦
ζ = lim
λ↑E min(λ) ∈ E ◦
By Transfer Principle, ζ ≤ ξ ≤ η is true for all ξ ∈ E ◦ , so that ζ and ξ are,
respectively, the minimum and the maximum of E ◦ .
We find opportune to give an explicit example of the property ruled by the
previous Proposition.
22

Example 1.7.13. Let E = R: by definition,
R ◦ = {ξ ∈ R ∗ : ξ = lim
λ↑Ω ϕ(λ), with ϕ(λ) ∈ λ}
1.7. EXTENSIONS OF SETS
From this set we are excluding, for instance, all the Ω-limits of functions that grows
“too fast”. If we call α = max(R ◦ ), then we have, for example, that α 2 ∈ R ◦ . In
this case, R ⊂ R ◦ ⊂ R ∗ , and all inclusions are strict.
On the other hand, it’s clear that R ∗ has no maximum and no minimum: for
every ϕ : Pfin(Ω) → R, it’s trivial to find ψ ′ and ψ ′′ : Pfin(Ω) → R satisfying the
inequalities
ψ ′ (λ) < ϕ(λ) < ψ ′′ (λ)
for all λ ∈ Pfin(Ω). Thanks to Transfer of inequalities, we deduce that R ∗ has no
maximum and no minimum.
Since the number α = max(R ◦ ) plays a central role in our theory, we find appropriate
to fix this notation through the rest of this paper with the following definition.
Definition 1.7.14. We define α = max(R ◦ ).
Another interesting property of hyperfinite sets is that every element of R ◦ other
than ±α has an “immediate predecessor” and an “immediate successor”. Formally,
this property can be written in the following way:
Proposition 1.7.15. If ξ ∈ R ◦ is different than ±α, there exists ξ + and ξ − ∈ R ◦
such that
1. ξ + > ξ and, if ζ ∈ R ◦ and ζ > ξ, then ζ ≥ ξ + ;
2. ξ − < ξ and, if ζ ∈ R ◦ and ζ < ξ, then ζ ≤ ξ − .
Proof. By hypothesis, there exists ϕ : Pfin(Ω) → R such that limλ↑R ϕ(λ) = ξ and
ϕ(λ ∩ R) ∈ λ ∩ R for all λ ∈ Ω. From this function, we can define
and
Sλ = {x ∈ R : x > ϕ(λ ∩ R)}
Iλ = {x ∈ R : x < ϕ(λ ∩ R)}
Notice that Sλ and Iλ are nonempty for a.e. λ ∈ Ω. If we call ψ + (λ) = min(λ ∩ Sλ)
and ψ − (λ) = max(λ ∩ Iλ) then, by the choiche of ξ, ψ ± are well defined for almost
every λ ∈ Ω. We can conclude that, if we call ξ ± = limλ↑R ψ ± (λ), then ξ ± ∈ R ◦ .
Now, we only need to show that ξ ± satisfy the thesis of the proposition. Notice
that, by construction and by the hypothesis ξ = ±α, the chain of inequalities
ψ + (λ ∩ R) > ϕ(λ ∩ R) > ψ − (λ ∩ R)
is true almost everywhere. Let’s now suppose that ζ ∈ R ◦ satisfies ζ > ξ. If we
write ζ = limλ↑R δ(λ), Transfer Principle ensures that δ(λ ∩ R) > ϕ(λ ∩ R) a.e. By
definition of ψ + , this also implies that δ(λ∩R) ≥ ψ + (λ∩R) holds almost everywhere,
so part 1 follows by an application of Transfer Principle. The second part of the
proposition can be proved in a similar way.
23

CHAPTER 1. Ω-CALCULUS
The properties ruled by Proposition 1.7.12 and by Proposition 1.7.15 can be
interpreted as particular instances of “Transfer Principle for sets”, where some class
of properties of finite sets are preserved under Ω-limit. This is the main reason of
our interest in hyperfinite extension of sets: they share some properties with finite
sets, and those properties may be used in interesting ways during the development
of calculus.
1.8 Extensions of functions
In the previous section, we have defined the natural and the hyperfinite extension
of subsets of R. Now, we wish to define a similar extension of functions with range
and domain contained in R.
Let’s consider a function f : A ⊆ R → R. By composing with f, we can turn
every function ϕ : Pfin(Ω) → A in a function f ◦ ϕ : Pfin(Ω) → R. Moreover, this
composition is coherent with the operation of Ω-limit.
Proposition 1.8.1. Let f : A ⊆ R → R and ϕ, ψ : Pfin(Ω) → A.
lim ϕ(λ) = lim ψ(λ) ⇒ lim f(ϕ(λ)) = lim f(ψ(λ))
λ↑Ω λ↑Ω λ↑Ω λ↑Ω
Proof. By definition, we have Qϕ,ψ ∈ Q and Qϕ,ψ = Qf(ϕ),f(ψ). The thesis follows
from Corollary 1.4.7.
Thanks to this result, we are allowed to define the extension of functions in
analogy to what we’ve done in the previous Section in the case of sets. In particular,
we will define two kind of extensions of functions: the natural extension and the
hyperfinite extension.
Definition 1.8.2. Let f : A ⊆ R → R. The natural extension of f, which we
denote f ∗ , is defined by:
f ∗
lim ϕ(λ)
λ↑Ω
= lim f(ϕ(λ))
λ↑Ω
Notice that, by definition, f ∗ : A ∗ → R ∗ .
The hyperfinite extension of a function f : A → R is defined as the restriction
of f ∗ to A ◦ :
f ◦ = f ∗
|A ◦ : A◦ → R ∗
Proposition 1.8.1 ensures that Definition 1.8.2 is well posed. Moreover, we have
that
∀a ∈ A, f ∗
(a) = f lim ca(λ)
λ↑Ω
= lim (f ◦ ca) (λ) = lim cf(a)(λ) = f(a)
λ↑Ω λ↑Ω
so the function f ∗ (and, as a consequence, the function f ◦ ), is an actual extension
of the function f.
Now, we want to show some properties of extension of functions. In the beginning,
we will focus on natural extension of functions because, by definition, the
24

1.8. EXTENSIONS OF FUNCTIONS
results will apply also to their hyperfinite extension. Then, in analogy to what happened
to the case of hyperfinite extension of sets, we will prove some properties of
hyperfinite extension of functions that are not satisfied by natural extension.
At first, we will show that the basic properties of functions are preserved under
natural and hyperfinite extension.
Proposition 1.8.3. Let f : A ⊆ R → B ⊆ R and g : B ⊆ R → C ⊆ R.
1. [f(A)] ∗ = f ∗ (A ∗ );
2. f is injective if and only if f ∗ is injective;
3. f is surjective if and only if f ∗ is surjective;
4. f is bijective if and only if f ∗ is bijective;
5. (g ◦ f) ∗ = g ∗ ◦ f ∗ .
Proof. To prove part 1, for a matter of commodity define B = [f(A)]. If ξ ∈ B ∗ ,
then ξ = limλ↑Ω ϕ(λ), where ϕ(λ) ∈ B for all λ ∈ Pfin(Ω). In particular, for every
λ, there is an element ψ(λ) ∈ A satisfying the equality f(ψ(λ)) = ϕ(λ). If we call
ζ = limλ↑Ω ψ(λ), then we have
f ∗ (ζ) = lim
λ↑Ω f(ψ(λ)) = lim
λ↑Ω ϕ(λ) = ξ
and this proves [f(A)] ∗ ⊆ f ∗ (A ∗ ). Conversely, if ζ = limλ↑Ω ψ(λ) for some ψ satisfying
ψ(λ) ∈ A for all λ ∈ Pfin(Ω), then f(ψ(λ)) ∈ B for all λ ∈ Pfin(Ω), hence
and this proves the other inclusion.
f ∗ (ζ) = lim
λ↑Ω f(ψ(λ)) ∈ B ∗
is injective. To prove the
For part 2, notice that if f ∗ is injective, then also f = f ∗
|A
other implication, suppose that there are ξ = limλ↑Ω ϕ(λ) and ζ = limλ↑Ω ψ(λ) satisfying
f ∗ (ξ) = f ∗ (ζ). This equality can be rewritten limλ↑Ω f(ϕ(λ)) = limλ↑Ω f(ψ(λ)).
By Transfer of equalities, we conclude f(ϕ(λ)) = f(ψ(λ)) a.e. and, by the hypothesis
that f is injective, we deduce ϕ(λ) = ψ(λ) almost everywhere. In particular,
this implies ξ = ζ.
Parts 3 and 4 of the Proposition follow from the equality stated in part 1.
To prove part 5, let ϕ : Pfin(Ω) → A. Then,
(g ◦ f) ∗
lim ϕ(λ)
λ↑Ω
= lim((g
◦ f) ◦ ϕ)(λ)
λ↑Ω
as we wanted.
= lim(g
◦ (f ◦ ϕ)(λ)
λ↑Ω
= g ∗
lim(f
◦ ϕ)(λ)
λ↑Ω
= g ∗
f ∗
lim ϕ(λ)
λ↑Ω
25

CHAPTER 1. Ω-CALCULUS
Corollary 1.8.4. Let f : A ⊆ R → B ⊆ R and g : B ⊆ R → C ⊆ R.
1. f is injective if and only if f ◦ is injective;
2. f is surjective if and only if f ◦ is surjective;
3. f is bijective if and only if f ◦ is bijective.
Proof. This is a consequence of Proposition 1.8.3 and the definition of f ◦ .
Notice that parts 1 and 5 of Proposition 1.8.3 don’t apply also to hyperfinite
extension of functions. We need to settle one more property of hyperfinite extension
of functions before we can show some explicit counterexamples. The property we
need is that monotony is preserved under natural and hyperfinite extension.
Proposition 1.8.5. Let f : A ⊆ R → R. f is strictly increasing if and only if f ∗
is strictly increasing. The same holds replacing “strictly increasing” with “strictly
decreasing”, “not increasing” and “not decreasing”.
Proof. Let’s suppose that f is strictly increasing, that is, x < y implies f(x) < f(y).
Let’s now consider ξ and ζ ∈ R ∗ , with ξ < ζ. If ξ = limλ↑Ω ϕ(λ) and ζ = limλ↑Ω ψ(λ),
by hypothesis we have Q ∋ {λ ∈ Pfin(Ω) : ϕ(λ) < ψ(λ)} = {λ ∈ Pfin(Ω) :
f(ϕ(λ)) < f(ψ(λ))}. Applying Transfer Principle, we conclude f(ξ) < f(ζ).
For the other parts of the proposition, the proof is similar: if f is not decreasing,
substitute “

1.9. TOWARDS A MODEL FOR Ω-CALCULUS
Example 1.8.8. Let A be the open set (0, 1) and f : A → R be the function
f(x) = 1/x 2 . Clearly, f(A) = (1, ∞), so [f(A)] ◦ = (1, ∞) ◦ . On the other hand, if
we define 1/β = min((0, 1) ◦ ), we know that there’s a function ϕ : Pfin(A) → R such
as limλ↑Ω ϕ(λ) = 1/β. If we compose ϕ with f ◦ , we deduce limλ↑A f ◦ (ϕ(λ)) = β 2 ,
and we don’t know if β 2 ∈ (1, ∞) ◦ .
In our theory, we have no way to show wether R ◦ is closed under square root or
wether β 2 ∈ (1, ∞) ◦ is true or false. However, we anticipate that, in Section 1.12,
we will show a model of Ω-Calculus where the above inclusion is false.
As for the case of composition of functions, since it’s generally false that f ◦ (A ◦ ) =
[f(A)] ◦ , the function g ◦ ◦ f ◦ may not even be well defined. On the other hand,
it’s always well defined the function g ∗ ◦ f ◦ , and the interested reader could easily
prove, with an argument similar to the proof of part 5 of Proposition 1.8.3, that
(g ◦ f) ◦ = g ∗ ◦ f ◦ .
1.9 Towards a model for Ω-Calculus
In the previous sections, we began to develop axiomatically a new mathematical
theory, without addressing the problem of its consistency. Now, we want to give a
proof of relative consistency for Ω-Calculus: in particular, we’ll show that Ω-Calculus
admits a model. This is a well known procedure, most notably used in the second half
of the XIX century to create euclidean models of non-euclidean geometries to show
that they were “at least as consistent as” euclidean geometry. For an introduction
about model theory and for a presentation of more recent applications, we remand
to [10] and to [11].
We begin by defining what a model for Ω-Calculus is.
Definition 1.9.1. A model for Ω-Calculus consists of a map
ϕ ↦→ lim
λ↑Ω ϕ(λ)
that assigns a Ω-limit limλ↑Ω ϕ(λ) to every function ϕ : Pfin(Ω) → R, in such a way
that the axioms of Ω-Calculus are realized.
In the next three sections, we will exhibit an algebraic model for Ω-Calculus:
in particular, every axiom of Ω-Calculus will correspond to an algebraic theorem,
whose truth can be proved by field theory. For this reason, we find useful to recall
the basic notions from Algebra that will be used during the construction of the
model. For an introduction to those topics, complete with proofs and details we
have omitted, see for instance [8] or [9].
Definition 1.9.2. Let A be a commutative ring with an identity element (from now
on, we will simply say “let A be a ring”, but we’ll also require that the ring is
commutative and has an identity element).
A set i ⊂ A is an ideal of A if it’s a subgroup of the additive group A and Ai = i
(e.g. for all a ∈ A and for all j ∈ i, aj ∈ i).
An ideal i ⊂ A is principal if there exists x ∈ i such as i = (x) (e.g. for all j ∈ i
exists a ∈ A satisfying ax = j).
27

CHAPTER 1. Ω-CALCULUS
An ideal i ⊂ A is prime if xy ∈ i implies x ∈ i or y ∈ i (or both).
An ideal i ⊂ A is maximal if for all ideals j ⊂ A, i ⊆ j implies i = j.
Definition 1.9.3. If i is an ideal of a ring A, we can form the quotient ring A/i.
First of all, we give an equivalence relation ≡i on A:
a ≡i b ⇔ a − b ∈ i
and then we consider the cosets [a]i = {x ∈ A : x − a ∈ i} induced by this relation.
If we call A/i = {[a]i : a ∈ A}, we can define a sum and a product on A/i based upon
those in A:
[a]i + [b]i = [a + b]i
[a]i · [b]i = [ab]i
It can be easily verified that these operations are well defined and independent of the
choice of representatives. With these operations, A/i has a ring structure inherited
from A.
If i is an ideal of A, we have the following characterizations: i is a prime ideal
if and only if A/i is an integral domain (that is, A/i has no zero divisors); i is a
maximal ideal if and only if A/i is a field. We can immediately deduce that every
maximal ideal is prime, because a field is an integral domain. The easy proof of
those statements can be found in [8] or in chapter 11 of [9].
Now, we’ll introduce a particular set, the set of real valued functions from
Pfin(Ω).
Fun(Pfin(Ω), R) = {ϕ : Pfin(Ω) → R}
This set has a natural structure of commutative ring with the following operations:
and
(ϕ + ψ)(λ) = ϕ(λ) + ψ(λ)
(ϕ · ψ)(λ) = ϕ(λ) · ψ(λ)
This ring is interesting to us, because we want to obtain a model for Ω-Calculus
from a a quotient of this ring. With this goal in mind, we find useful to show some
examples of ideals in Fun(Pfin(Ω), R).
Example 1.9.4. Let λ0 ∈ Pfin(Ω) and iλ0 = {ϕ : Pfin(Ω) → R : ϕ(λ0) = 0}. We
will show that iλ0 is a maximal, principal ideal of A = Fun(Pfin(Ω), R).
Clearly, iλ0 is an additive subgroup of A. Now, let’s choose once and for all
ϕ ∈ iλ0 such as ϕ(λ) = 0 whenever λ = λ0. We have that, for all η ∈ A
(η · ϕ)(λ0) = η(λ0) · ϕ(λ0) = 0
so that the inclusion ϕA ⊆ iλ0 is true. This shows also that Aiλ0 = iλ0, from which
we can conclude that iλ0 is indeed an ideal in A.
Now, we want to show that iλ0 = (ϕ) or, in other words, that every function
ψ ∈ iλ0 can be written as ϕ · η for a suitable η ∈ A. Let’s pick ψ ∈ iλ0 and define a
new function
ψ(λ)/ϕ(λ) if λ = λ0
η(λ) =
1 if λ = λ0
28

1.9. TOWARDS A MODEL FOR Ω-CALCULUS
By definition, η ∈ A and (ϕ · η)(λ) = ψ(λ) for all λ ∈ Pfin(Ω). This, combined with
the inclusion we have shown above, allows to conclude iλ0 = (ϕ).
To see that iλ0 is maximal, we will prove that Fun(Pfin(Ω), R)/iλ0 is a field. Let’s
choose [ϑ] ∈ Fun(Pfin(Ω), R)/iλ0 with ϑ ∈ iλ0. We want to show that we can find an
invertible representative for [ϑ].
If ϑ(λ) = 0 for all λ ∈ Pfin(Ω), then is invertible in Fun(Pfin(Ω), R) (its inverse
being the well defined function 1/ϑ), and then, passing to the quotient map, we see
that [1/ϑ] is the inverse of [ϑ] in Fun(Pfin(Ω), R)/iλ0.
Let’s now suppose that Qϑ,c0 = ∅ (e.g. ϑ(λ) = 0 for some λ ∈ Pfin(Ω)). Since
[ϑ] = 0 in Fun(Pfin(Ω), R)/iλ0, we can assume that ϑ(λ0) = ξ = 0. If we define the
function
ϑ ′
ξ if λ = λ0
(λ) =
ϑ(λ) 2 + 1 if λ = λ0
we have that ϑ − ϑ ′ ∈ iλ0 and this, by definition, means that [ϑ ′ ] = [ϑ]. But it’s
easily seen that ϑ ′ (λ) = 0 for all λ ∈ Pfin(Ω), and so ϑ ′ is invertible, and so it is
[ϑ ′ ]. This shows that [ϑ] has an invertible representative, as we wanted.
One of the most important ideals that will be useful in giving a model for Ω-
Calculus is the ideal of the functions eventually equal to zero.
Definition 1.9.5. Let’s call
i0 = {ϕ ∈ Fun(Pfin(Ω), R) : ∃λ0 ∈ Pfin(Ω) such as ϕ(λ) = 0 ∀λ ⊇ λ0}
the set of all functions eventually equal to zero.
Proposition 1.9.6. The set i0 is a nonprincipal ideal of Fun(Pfin(Ω), R), and is
not prime (and, therefore, is also not maximal).
Proof. We omit the routine checkings of the fact that i0 is an ideal.
To see that i0 is nonprincipal, we want to show that, for every ϕ ∈ i0, there exists
ψ ∈ i0 such that, for all η ∈ Fun(Pfin(Ω), R), ϕ · η = ψ. So let’s choose ϕ ∈ i0. By
definition of i0, there exists λ0 ∈ Pfin(Ω) such as ϕ(λ) = 0 for all λ ⊇ λ0. Now, let
λ1 ⊃ λ0. If we define
1 if λ ⊆ λ1
ψ(λ) =
0 if λ ⊃ λ1
it’s easily checked that ψ ∈ i0. What we want to show is that, for all η ∈
Fun(Pfin(Ω), R), ϕ · η = ψ. But this is easily seen: for all η ∈ Fun(Pfin(Ω), R),
the following relation holds:
(ϕ · η)(λ1) = ϕ(λ1) · η(λ1) = 0 · η(λ1) = 0
while, on the other hand, ψ(λ1) = 1.
To see that i0 is not prime, we will find ϕ and ψ ∈ Fun(Pfin(Ω), R) such as
ϕ, ψ ∈ i0, but ϕ · ψ ∈ i0. In order to achieve this goal, we need to construct a strictly
increasing sequence of subsets in Pfin(Ω).
First of all, let’s choose λ0 ∈ Pfin(Ω). It’s easy to see that the set A0 =
{λ ∈ Pfin(Ω) : λ ⊃ λ0} is nonempty, so we can choose λ1 ∈ A0. In the same spirit,
29

CHAPTER 1. Ω-CALCULUS
we define A1 = {λ ∈ Pfin(Ω) : λ ⊃ λ1} and, once again, A1 = ∅. As we did before,
we can pick an arbitrary λ2 ∈ A2: proceding this way, we obtain a sequence {λi} i∈N
such as λi ⊂ λj whenever i < j.
Now, we define ϕ and ψ in the following ways:
1 if λ ∈ {λi}
ϕ(λ) =
i∈N
0 if λ ∈ {λi} i∈N
and
0 if λ ∈ {λi}
ψ(λ) =
i∈N
1 if λ ∈ {λi} i∈N
It’s clear that both ϕ and ψ are not eventually equal to zero, but their product is
c0, and c0 ∈ i0.
It’s a straightforward application of Zorn’s Lemma (an equivalent formulation of
the Axiom of Choice) to see that
Proposition 1.9.7. If i ⊂ A is an ideal of A, there exists a maximal ideal of A
containing i.
The proof, that requires Zorn’s Lemma, an equivalent formulation of the Axiom
of Choice, can be found on every algebra textbook (once again we remand to [8] or
to Theorem 11.17 of [9]).
Proposition 1.9.7 ensures that there is a maximal ideal m ⊃ i0. Moreover, this
ideal is not principal.
Proposition 1.9.8. If m is a maximal ideal containing i0, then m is not principal.
Proof. Suppose m is principal, so that there is ϕ ∈ m satisfying m = (ϕ). Since m
is an ideal, m = Fun(Pfin(Ω), R) and so ϕ is not invertible. This implies that there
is λ0 ∈ Pfin(Ω) such as ϕ(λ0) = 0.
If we define a function
1 if λ ⊆ λ0
ψ(λ) =
0 if λ ⊃ λ0
we have clearly ψ ∈ i0. Since for all η ∈ Fun(Pfin(Ω), R), (ϕ · η)(λ0) = 0, we
can conclude that ψ = ϕ · η for all η ∈ Fun(Pfin(Ω), R), but this contradicts the
hypothesis that m is principal. We conclude that m is not principal.
Combining Propositions 1.9.7 and 1.9.8, we can deduce the existence of a nonprincipal
maximal ideal in the ring Fun(Pfin(Ω), R). Moreover, this ideal can be
characterized as a maximal ideal extending i0. This allows us to show that, in order
to have a model in which Axioms 1, 2 and 3 of Ω-Calculus hold, it’s sufficient to
quotient Fun(Pfin(Ω), R) with a nonprincipal maximal ideal.
Theorem 1.9.9. Let m be a nonprincipal maximal ideal of the ring Fun(Pfin(Ω), R)
and let π : Fun(Pfin(Ω), R) → Fun(Pfin(Ω), R)/m the canonical projection
π(ϕ) = [ϕ]m
where we agree that cosets [cr]m of constant sequences are identified with the corresponding
real numbers. Then, Axioms 1, 2 and 3 of Ω-Calculus are satisfied in
Fun(Pfin(Ω), R)/m.
30

1.10. FILTERS AND ULTRAFILTERS
Proof. First of all, we observe that
R ∗
= lim ϕ(λ) : λ : Pfin(Ω) → R
λ↑Ω
= Fun(Pfin(Ω), R)/m
and Fun(Pfin(Ω), R)/m is a field because m is maximal. This is enough to satisfy
Axiom 1.
Axiom 2 is verified, since we identified limλ↑Ω cr = [cr]m with r ∈ R.
By construction, sums and products in Fun(Pfin(Ω), R)/m commute with cosets,
so that Axiom 3 is verified.
1.10 Filters and ultrafilters
Constructing a model in which Numerosity Axiom 1 holds is a little trickier. We
can’t achieve this resulyt by using maximal ideals alone, but we’ll need to use another
algebraic tool, namely filters and ultrafilters.
Definition 1.10.1. A filter F on a set A is a nonempty family of subsets of A
satisfying
1. ∅ ∈ F
2. if X ∈ F and Y ⊇ X, then Y ∈ F
3. if X, Y ∈ F, then X ∩ Y ∈ F
Remark 1.10.2. We want to observe that the notion of a filter F on a set A can
be reinterpreted with the language of algebra. Consider the set P(A) with the
two operations ∩ and ∪: we can think of ∩ as the “sum” in P(A) and ∪ as the
“product”. With this interpretation, we have that A is the identity element of ∩
and the absorbing element of ∪ (like 0 ∈ Z is the identity element of the sum and
the absorbing element of product) and ∅ is the identity element of ∪. It should be
clear that P(A), endowed with the operations ∩ and ∪, becomes a commutative
ring. Whenever we will talk about this ring, we will denote by 0A and 1A the sets
A and ∅ respectively.
A filter F on A can then be interpreted as an ideal of the ring P(A): condition 1
of Definition 1.10.1 says that 1A ∈ F, condition 2 says that AF = F and condition 3
says that F is closed under sums and hence a subgroup of the additive group P(A).
Definition 1.10.3. We say that a family of sets F has the finite intersection property
(FIP) if
k
∀X1 . . . Xk ∈ F, Xk = ∅
With the interpretation of Remark 1.10.2, we say that F has the FIP if the
additive subgroup of A generated by F does not contain 1A (hence is different from
the whole P(A)).
i=1
31

CHAPTER 1. Ω-CALCULUS
Definition 1.10.4. If G has the FIP, then
〈G〉 =
is the filter generated by G.
Y : ∃X1, . . . , Xk ∈ G so that Y ⊇
The name “filter generated by G” is justified by the fact that 〈G〉 is indeed a
filter, and it’s the smallest filter containing G. We won’t give the easy proof of this
statement.
Example 1.10.5. For all X ⊆ A, FX = {Y ⊆ A : Y ⊇ X} is a filter. The filters of
the form FX with X ⊆ A are called principal filters. It’s clear that all the principal
filters have the FIP.
With the interpretation of Remark 1.10.2, a principal filter FX corresponds to
the (principal) ideal generated by X.
It’s important not to think all filters are principal. In fact there is one example
of an important nonprincipal filter: the Frechet filter.
Definition 1.10.6. The Frechet filter Fr(A) over a set A is the family of cofinite
subsets of A:
Fr(A) = {X ⊆ A : X c is finite}
This filter will play an important role in the construction of models for Ω-
Calculus.
In analogy to the case of ideals, we can define filters that are maximal with respect
to inclusion. This condition turns out to be equivalent to other two characterizations.
Proposition 1.10.7. Let F be a filter over a set A. The following properties are
equivalent:
1. X ∈ F ⇒ X c ∈ F;
2. k
i=1 Xk ∈ F ⇒ ∃i : Xi ∈ F;
3. F is a maximal filter on A with respect to inclusion.
Proof. All the proofs are by reductio ad absurdum.
To prove that condition 1 implies condition 2, suppose that condition 1 holds and
that k i=1 Xk ∈ F, but Xi ∈ F for all i = 1, . . . , k. Then, by hypothesis, Xc i ∈ F
for all i = 1, . . . , k; and the same holds for the intersection
k
X c c k
i =
∈ F
i=1
i=1
Since we are assuming that k
i=1 Xk ∈ F, this is a contradiction: otherwise, we
would deduce that ∅ ∈ F, and conclude that F is not a filter.
Now, we will prove that condition 2 implies condition 3. If F was not maximal,
then there would be a filter F ′ that properly includes it. Now, pick any X ∈ F ′ \ F:
32
Xk
k
i=1
Xk

1.10. FILTERS AND ULTRAFILTERS
clearly, X c ∈ F ′ and, as a consequence, X c ∈ F. We conclude that A = X ∪X c ∈ F
is the union of two sets neither belonging to F, and this is against the hypothesis.
In order to prove that condition 3 implies condition 1, suppose that, for some
set X, both X and X c ∈ F. We want to show that the family
G = {Y ∩ X : Y ∈ F}
has the FIP. Notice that, by definition Y ∩ X = ∅ for all Y ∈ F: otherwise, we
would deduce Y ⊆ X c , and this would imply X c ∈ F against the hypothesis. Now,
suppose that (Y1 ∩X), . . . , (Yk ∩X) ∈ G: then, Y1, . . . , Yk ∈ F, and also k
i=1 Yi ∈ F.
Since
(Y1 ∩ X) ∩ . . . ∩ (Yk ∩ X) =
k
we deduce that G has the FIP. Now, if we consider the filter F ′ = 〈G〉, we claim
that F ′ properly includes F. In fact, for every Y ∈ F, Y ⊇ Y ∩ X, and so Y ∈ F ′ .
Moreover, X ∈ F but X = A∩X ∈ F ′ . This is in contradiction with the maximality
of F.
Definition 1.10.8. A filter satisfying one (hence all) of the properties itemized in
Proposition 1.10.7 is called an ultrafilter.
We want to explicitely observe that, if A is infinite, the Frechet filter Fr(A) is
not an ultrafilter. In fact, we can easily find a set X ⊆ A such that both X and X c
are infinite. It’s clear that X ∪ X c = A ∈ Fr(A), while both X, X c ∈ Fr(A).
Now, we want to give a characterization of principal ultrafilters.
Proposition 1.10.9. A principal filter FX is an ultrafilter if and only if X = {a}
is a singleton.
Proof. If X contains at least two elements, we can partition X = X1 ∪ X2 into two
disjoint nonempty set. By the property characterizing ultrafilters, we would have
that either X1 ∈ FX, or X2 ∈ FX. Both cases lead to a contradiction, because X1
and X2 are proper subsets of X, while FX only contains supersets of X. The reverse
implication is trivial, because
i=1
Yk
∩ X
X ∈ F{a} ⇔ a ∈ X ⇔ a ∈ X c ⇔ X c ∈ F{a}
We will call Ua the principal ultrafilters generated by {a} and, with abuse of
notation, we will say that Ua is the ultrafilter generated by a.
Now, we proceed by giving a characterization of ultrafilters over finite sets.
Proposition 1.10.10. All ultrafilters on finite sets are principal.
Proof. Let U be an ultrafilter on a finite set A. Then A = {a1}∪. . .∪{ak} ∈ U and,
using the property of ultrafilters, we deduce that one of the {ai} ∈ U: we know that
{a1} ∪ ({a2} ∪ . . . ∪ {ak}) ∈ U, and this implies {a1} ∈ U or {a2} ∪ . . . ∪ {ak} ∈ U. In
the first case, we’ve concluded. Otherwise, we can proceed in a similar way, until,
by the finiteness of A, we find some {ai} ∈ U. If there was j = i such that {aj} ∈ U,
this would imply {ai} ∩ {aj} = ∅ ∈ U, in contradiction with the fact that U is a
filter. We conclude U = Uai .
33

CHAPTER 1. Ω-CALCULUS
In the same spirit, we can give a characterization of nonprincipal ultrafilters over
infinite sets.
Proposition 1.10.11. An ultrafilter on an infinite set A is nonprincipal if and only
if it contains no finite sets, hence if and only if it contains the Frechet filter Fr(A).
Proof. Let’s suppose Fr(A) ⊂ U. This implies that, for all a ∈ A, {a} c ∈ U, and so
{a} ∈ U, from which we conclude that U is not principal.
Now, let’s suppose that Fr(A) ⊆ U. By definition, we can find a cofinite set
X ∈ U. Since U is an ultrafilter, we have that X c = {a1} ∪ . . . ∪ {ak} ∈ U and then,
applying the argument of proof of Proposition 1.10.10, we conclude that {ai} ∈ U
for exactly one i. We conclude that U = Uai is principal, and so the proof is
concluded.
It’s time to settle a result about the existence of ultrafilters. In analogy to the
algebraic proof of the existence of maximal ideals, this proof is a standard reasoning
involving Zorn’s Lemma, so we will omit some simple verifications.
Proposition 1.10.12. Every filter F over a set A can be extended to an ultrafilter.
Proof. Let’s consider the set
G = {G filter on A : G ⊇ F}
with the partial order given by inclusion.
It’s straightforward to verify that every chain in G has an upper bound (just
take the union of all filters in the chain and verify it’s a filter containing F). We can
apply Zorn’s Lemma, that guarantees the existence of a maximal element U ∈ G.
Clearly, U ⊇ F, and it’s easy to verify that U is an ultrafilter.
As a consequence, we obtain the following
Corollary 1.10.13. For every infinite set A, there exist a nonprincipal ultrafilters
on A.
Proof. Proposition 1.10.11 ensures that every ultrafilter containing the Frechet filter
is nonprincipal. The existence of such ultrafilters is guaranteed by Proposition
1.10.12.
The importance of ultrafilters is that there’s a relation between nonprincipal
maximal ideals in the ring Fun(Pfin(Ω), R) and nonprincipal ultrafilters over the
set Pfin(Ω). This is a consequence of a more general fact, that we’ll show before
focusing on the case of our interest. We want to warn the reader that, with abuse
of notation, if ϕ ∈ Fun(A, B), we will use the notation Qϕ,c0 to denote the set
{a ∈ A : ϕ(a) = 0}.
Proposition 1.10.14. If Fun(A, B) with the two operations
34
(ϕ + ψ)(a) = ϕ(a) + ψ(a)
(ϕ · ψ)(a) = ϕ(a) · ψ(a)

is a commutative ring and F is an ultrafilter on A, then
mF = {ϕ ∈ Fun(A, B) : Qϕ,c0 ∈ F}
is a maximal ideal of Fun(A, B).
Conversely, if m is a maximal ideal of Fun(A, B), then
Fm = {Qϕ,c0 : ϕ ∈ m}
1.10. FILTERS AND ULTRAFILTERS
is an ultrafilter on A.
The two operations F ↦→ mF and m ↦→ Fm are one the inverse of the other, and
in this corrispondence the principal (resp. non principal) ultrafilters corresponds to
the principal (resp. non principal) maximal ideals.
Proof. The identities FmF = F and mFm can be easily checked using the definitions.
Let m be a maximal ideal of Fun(A, B) and Fm the corresponding family of sets.
Since for all ϕ, ψ ∈ Fun(A, B) we have Qϕ+ψ,c0 ⊇ Qϕ,c0 ∩ Qψ,c0, the fact that m is
closed under sums implies that Fm is closed under finite intersections. In the same
spirit, if ϕ ∈ m and ψ ∈ Fun(A, B), then Qϕ·ψ,c0 ⊇ Qϕ,c0, and this, coupled with the
fact that Fun(A, B)m = m, implies that Fm is closed under superset. Now, choose
X ⊆ A in a way that X ∈ Fm, and then let χ be its characteristic function. Clearly,
χ · (1 − χ) = c0 ∈ m and, because m is prime, χ ∈ m or 1 − χ ∈ m. But the latter
is false by definition, since Q1−χ,c0 = X ∈ Fm. We conclude χ ∈ m, and this implies
Qχ,c0 = Xc ∈ Fm. This and the fact that c0 ∈ m ensures that ∅ ∈ Fm, and so Fm is
an ultrafilter.
Conversely, let F be an ultrafilter on A. To see that mF is a proper subset of
Fun(A, B), consider that Qc1,c0 = ∅ and, since ∅ ∈ F, c1 ∈ mF, so mF = Fun(A, B).
Now, suppose that ϕ, ψ ∈ mF: we have already seen that Qϕ+ψ,c0 ⊇ Qϕ,c0 ∩ Qψ,c0
and, since F is a filter, then Qϕ,c0 ∩ Qψ,c0 ∈ F, so we can conclude ϕ + ψ ∈ mF. In
the same spirit, for all ϕ ∈ mF and for all ψ ∈ Fun(A, B), we deduce Qϕ·ψ,c0 ∈ F,
and this implies Fun(A, B)mF = mF, completing the proof that mF is an ideal.
Now, by contradiction, suppose that mF is not maximal: then, by the first part
of the proposition, we would have that FmF
= F is not an ultrafilter, against the
hypothesis. We can conclude that mF is a maximal ideal.
It can be directly verified that, for every a ∈ A, Fm is the principal ultrafilter
generated by a if and only if mF is the principal maximal ideal generated by a.
As a consequence of the previous Proposition, if we take A = Pfin(Ω) and B = R,
we obtain immediately the following result:
Proposition 1.10.15. If F is an ultrafilter on Pfin(Ω), then
mF = {ϕ ∈ Fun(Pfin(Ω), R) : Qϕ,c0 ∈ F}
is a maximal ideal on Fun(Pfin(Ω), R).
Conversely, if m is a maximal ideal of Fun(Pfin(Ω), R), then
is an ultrafilter on Pfin(Ω).
Fm = {Qϕ,c0 : ϕ ∈ m}
35

CHAPTER 1. Ω-CALCULUS
The two operations F ↦→ mF and m ↦→ Fm are one the inverse of the other, and
in this corrispondence the principal (resp. non principal) ultrafilters corresponds to
the principal (resp. non principal) maximal ideals.
We are interested in ultrafilters that are in correspondence with maximal ideals
m ⊃ i0. We explicitely observe that those ultrafilters have an additional, interesting
property: they are fine.
Definition 1.10.16. A filter F on Pfin(A) is fine if, for all a ∈ A, {X ∈ Pfin(A) :
{a} ⊆ X} ∈ F.
This property has particular importance, since the proof of part 2 of Theorem
1.9.9 depends on the identification of eventually equal functions, and this identification
is possible because the nonprincipal maximal ideals extend i0. This hypothesis
is essential in order to have a model in which Axiom 2 of Ω-Calculus holds.
1.11 A model for Ω-Calculus
Thanks to the results of the previous section, we can focus in constructing a nonprincipal
fine ultrafilter on Pfin(Ω) in a way that Numerosity Axiom 1 holds. Then,
we’ll use the corrispondence of Proposition 1.10.15 to obtain a nonprincipal maximal
ideal that will give rise to a model of Ω-Calculus in which Numerosity Axiom 1
holds.
We start by constructing a particular family of sets ΛR ⊂ Pfin(R), which will be
essential to define the desired ultrafilter. In order to define ΛR, we will start with a
subfamily ΛQ, and then we’ll define ΛR in a way that ΛQ = ΛR ∩ P(Q).
We begin by posing
p
λn = : p ∈ Z, |p| ≤ n2
n
or, more explicitely,
λn = −n, − n2 − 1 1 1
, . . . , − , 0,
n n n , . . . , n2
− 1
, n
n
Now, we define the family ΛQ in the following way:
ΛQ = {λn : n = m!, m ∈ N}
We remark that the restriction n = m! ensures that, for every λn ′, λn ′′ ∈ ΛQ, n ′ ≤ n ′′
implies λn ′ ⊆ λn ′′, and so ⊆ is a total order on ΛQ.
Before we define ΛR, we need to construct another family of sets: we call
Θ = Pfin([0, 1] \ Q)
In other words, Θ is the family of finite sets of irrational numbers between 0 and 1.
Now, for all λn ∈ ΛQ and θ ∈ Θ, we set
p + a
λn,θ = λn ∪ :
n
p
n ∈ λn
\ {n}, a ∈ θ
36

1.11. A MODEL FOR Ω-CALCULUS
We can think that λn,θ is constructed in the following way: we started with the
segment [−n, n] and divided it in n 2 parts of lenght 1/n, so we got a subdivision
into segments of the form p/n, (p + 1)/n , where p = −n 2 , −n 2 + 1, . . . , n 2 . In each
of those segments, we put a “rescaled” copy of θ. Thus, the set λn,θ contains n 2 + 1
rational points and n 2 · |θ| irrational points.
Finally, we set
ΛR = {λn,θ : n = m!, m ∈ N, θ ∈ Θ}
Now, we want to show that the numerosity axiom holds or, in a more formal
way:
Proposition 1.11.1. If a, b ∈ Q,
lim
λn,θ↑R |[a, b) ∩ λn,θ| = (b − a) · n([0, 1))
Proof. Let’s take an interval [a, b) where a, b ∈ Q. If n is sufficiently large, [a, b)∩λn,θ
contains n(b − a) rational numbers and n|θ|(b − a) irrational numbers, so that
Then,
|[a, b) ∩ λn,θ| = n(b − a)(|θ| + 1)
n([a, b)) = lim
λn,θ↑R |[a, b) ∩ λn,θ| = lim |n(b − a)(|θ| + 1)|
λn,θ↑R
= (b − a) · lim n(|θ| + 1) = (b − a) · n([0, 1))
λn,θ↑R
So far, we have built a family of sets ΛR in a way that, if we take the limit with
respect to the sets of that family, the numerosity axiom holds. Now, we want to
create an ultrafilter UR over Pfin(Ω) that contains all elements of ΛR. Let’s pause
for a moment and think about what properties we want UR to satisfy:
1. UR must be nonprincipal and fine
2. Since the function |[a, b) ∩ λ| is mapped to its limit limλ↑Ω |λ ∩ [a, b)|, we want
this limit to be equal to limλn,θ↑Ω |[a, b) ∩ λn,θ|
To satisfy the second condition, it is sufficient that the function |[a, b) ∩ λ| is in
the same equivalence class of |[a, b) ∩ λn,θ| modulo the nonprincipal maximal ideal
we want to determine. A little thought shows that it’s enough that UR contains the
filter FΛR = {Λ ⊆ Pfin(Ω) : Λ ⊇ ΛR}. We want to explicitely note that, if Λ ∈ FΛR ,
then Λ is infinite (because ΛR is).
Putting everything together, we conclude that, in order to obtain our model, we
need to find a nonprincipal fine ultrafilter containing FΛR .
Before we achieve this result, we will prove that the family FΛR ∪ Fr(Pfin(Ω))
has the FIP.
Lemma 1.11.2. FΛR ∪ Fr(Pfin(Ω)) has the FIP.
37

CHAPTER 1. Ω-CALCULUS
Proof. Since both FΛR and Fr(Pfin(Ω)) have the FIP, it’s sufficient to prove that,
for all Λ ∈ FΛR and for all X ∈ Fr(Pfin(Ω)), Λ ∩ X = ∅. But this is readily seen:
since X c is finite and Λ ⊇ ΛR is infinite, Λ ⊆ X c , hence Λ ∩ X = ∅.
As a consequence of this Lemma, we deduce that 〈FΛR ∪ Fr(Pfin(Ω))〉 is a well
defined filter. This fact will be useful during the proof of the next Proposition.
Proposition 1.11.3. There exists a nonprincipal fine ultrafilter on Pfin(Ω) that
extends FΛR .
Proof. We will modify the proof of 1.10.12 to obtain the desired ultrafilter.
Let’s consider the set
G = {G filter on Pfin(Ω) : G ⊇ FΛR
and G is fine and contains no finite set}
with the partial order given by inclusion. If we prove that G = ∅, then the rest
of the proof is a straightforward application of Zorn’s Lemma. But, thanks to
Lemma 1.11.2, we can immediately see that G contains the filter generated by FΛR ∪
Fr(Pfin(Ω)), so G = ∅.
Once again, we leave to the reader the checkings that every chain in G has an
upper bound (as in the proof of Proposition 1.10.12, take the union of all filters in
the chain and verify it’s a fine filter that contains no finite sets and extends FΛR ).
We can apply Zorn’s Lemma and obtain the ultrafilter UR ∈ G.
Observe that, since UR ∈ G, UR is fine and contains no finite set. By Proposition
1.10.11, this and the fact that Pfin(Ω) is infinite imply that UR is nonprincipal.
Finally, we can apply Proposition 1.10.15 to UR and find the corresponding nonprincipal
maximal ideal mR. Now, we will prove that Fun(Pfin(Ω), R)/mR is a model
for Ω-Calculus.
Theorem 1.11.4. Let mR be the nonprincipal maximal ideal defined above and let
π : Fun(Pfin(Ω), R) → Fun(Pfin(Ω), R)/mR the canonical projection
π(ϕ) = [ϕ]mR
where we agree that cosets [cr]mR of constant sequences are identified with the corresponding
real numbers. Then, all the axioms of Ω-Calculus are satisfied.
Proof. Theorem 1.9.9 ensures that Axioms 1 to 3 are verified.
Propositions 1.11.1 and 1.11.3 ensure that the function |[a, b) ∩ λ| is mapped to
(b − a) if a, b ∈ Q, and so Numerosity Axiom 1 holds as well.
This shows that Ω-Calculus has a model.
Before we move on, we want to observe that this process of model building is
general and can be applied to other situations. If we want to develop a theory based
on the analogue of Axioms 1, 2 and 3, we can build a model in a manner similar
to the one shown in this section. Moreover, we can even change domain D and
codomain C for our functions, as long as Fun(D, C) is a commutative ring.
We conclude by anticipating that, in Section 4.5, we will generalize further this
model building method to give a model for an extension of the theory presented in
this chapter.
38

1.12. REASONING IN THE MODEL
Remark 1.11.5. We want to note explicitely that, in this model we’ve built, the
family of qualified sets Q of Definition 1.4.3 is nothing but our ultrafilter UR. What
we’ve said in Remark 1.4.5 correspond precisely to the idea that the Ω-limit of
functions is defined modulo being equal in a qualified set.
This example suggests a generalized notion of limit, where we consider an arbitrary
filter over a set I. Precisely, suppose that {xi}i∈I is an I-sequence of elements
in a given topological space X, and F is a filter on I. We can define the F -limit of
{xi}i∈I by setting:
F − lim xi = x ⇔ {i ∈ I : xi ∈ A} ∈ F for all neighborhoods A of x
It turns out that, with some hypothesis over X, the notion of F -limit has some
“good” properties. Since the study of this generalization of the concept of limit is
beyond the scope of this paper, for further references we remand to Chapter 7 of [1].
1.12 Reasoning in the model
The existence of a model for Ω-Calculus has a number of interesting consequences
that go way beyond the consistency of the theory. In particular, having explicitely
determined a family of “qualified” sets, namely the family ΛR ⊂ UR, allows us to
explicitely calculate some Ω-limits that were otherwise impossible to determine. As a
consequence, by reasoning in the model of Ω-Calculus, we can show some interesting
results whose validity depend on the ultrafilter UR.
We begin by showing some properties of the number α.
Proposition 1.12.1. In our model, α = max(N ◦ ). Moreover, for all k ∈ N, α is a
multiple of k.
Proof. We recall that α = max(R ◦ ) and, in our model,
α = max(R ◦ ) = lim
λ↑Ω max(λn,θ) = lim
λ↑Ω n!
Since, by definition, n! ∈ N, we conclude that α ∈ N ◦ . Moreover, if we pick k ∈ N,
we have that n! is eventually multiple of k, and the second part of the thesis follows
by applying Transfer Principle.
This property has some interesting consequences. First of all, it shows that, by
an appropriate choiche of the model of Ω-Calculus and thanks to Transfer Principle,
we can find a model where α satisfies any finite number of elementary properties. If
we needed, for instance, that n√ α ∈ N ∗ for all n ∈ N, we would have posed
ΛQ = λn : n = m k , m, k ∈ N
Notice that this new definition of ΛQ would have determined a different choice of
the ultrafilter UR.
A second consequence of Proposition 1.12.1 is that there is a relation between α
and n(N), as shown by the following example.
39

CHAPTER 1. Ω-CALCULUS
Example 1.12.2. We want to calculate n(N). To do so, we use that, in our model,
limλ↑N |λ| = limλn,θ↑N |λn,θ|. Since
and
we deduce that, by definition of α,
λn,θ ∩ N = {1, . . . , n}
n = |λn,θ ∩ N| = max(λn,θ)
n(N) = lim
λ↑N |λ| = lim
λn,θ↑R max(λn,θ) = α (1.4)
Remark 1.12.3. Equation 1.4 shows us another interesting property of the number
α:
α = lim
λ↑R max(λ) = lim
λ↑N |λ|
We explicitely remark that this equality is true in the model, but is not entailed by
any axiom of Ω-Calculus. In other words: this equality is generally false, unless we
require that it holds by adding an appropriate axiom or by choosing a particular
model of Ω-Calculus.
We continue by calculating the numerosities of the sets E and O of even and odd
numbers, and those of Z and Q.
Example 1.12.4. We have already observed that, since N = E ∪ O, then, by finite
additivity of numerosity, n(N) = n(E) + n(O). Now, we will determine n(E) and
n(O). As in the previous example, n(E) = limλn,θ↑E |λn,θ|. If n > 1, then
|λn,θ ∩ E| = {2, . . . , n}
from which we deduce |λn,θ ∩ E| = n/2 is eventually true. We conclude n(E) = α/2
and n(O) = n(N) − n(E) = α/2.
Example 1.12.5. We want to calculate n(Z). Since Z = N ∪ −N ∪ {0}, by using
Corollary 1.6.3, we conclude n(Z) = 2n(N) + 1 = 2α + 1.
Example 1.12.6. We want to calculate n(Q). As in Example 1.12.2, we have
limλ↑Q |λ| = limλn,θ↑R |λn,θ ∩ Q|. We recall that, by definition,
λn,θ ∩ Q = λn = −n, −n2 + 1
, . . . , −
n
1 1
, 0,
n n , . . . , n2
− 1
, n
n
so |λn| = 2n 2 + 1. Taking the Ω-limit, we obtain n(Q) = 2α 2 + 1.
In our model, we can also answer the open question of Example 1.8.8. In particular,
we will show that min((0, 1) ◦ ) < 1/α.
Proposition 1.12.7. min((0, 1) ◦ ) < 1/α
40

1.12. REASONING IN THE MODEL
Proof. By the definition of the ultrafilter UR, we have that, if we define xn,θ =
min(λn,θ ∩ (0, 1)), then xn,θ is an irrational number satisfying the inequalities 0 <
xn,θ < 1/n, where n = m! ∈ N. By Transfer of inequalities, the chain of inequalities
holds for the Ω-limits
0 < lim xn,θ < lim
λ↑Ω λ↑Ω
from which we immediately conclude.
1
max(λn,θ)
= 1
α
Thanks to this result, we can go back to Example 1.8.8 and conclude the reasoning.
Since f|(0,1) is strictly decreasing, then so is f ◦ by Proposition 1.8.5. Thanks
to Proposition 1.12.7, we know that β = min((0, 1) ◦ ) < 1/α, so we can conclude
f ◦ (1/β) > f ◦ (1/α) = α 2 . Since α 2 ∈ (1, +∞) ◦ , we deduce that also f ◦ (β) ∈
(1, +∞) ◦ .
We explicitely remark that, had we chosen another ultrafilter, some (or maybe
all!) of these results could have been different. From now on, we will choose this
model for Ω-Calculus and we will feel free to use some of its peculiar properties for
further developments of the theory.
Remark 1.12.8. We observe that, in the proof of Proposition 1.12.1, we wrote an
expression of the form limλ↑Ω f(n). The meaning of this writing can be rendered
precise in a number of ways, one of them being
In our model, we could have also used
lim f(n) = lim f(|λ|)
λ↑Ω λ↑Ω
lim f(n) = lim
λ↑Ω λn,θ↑R f(max(λn,θ))
as we did in the proof of Proposition 1.12.1. Observe that, in our model, max(λn,θ)
is indeed a natural number and, since {λn,θ}n∈N ∈ Q, this limit is well defined. In
the future, when the need arises, we will choose whatever way we will think more
appropriate for the specific situation.
We conclude this chapter by showing that there are some relations between
Ω-Calculus and Alpha-Calculus, a similar theory that rules the “Alpha-limit” of
sequences instead of more general functions. For a complete presentation of this
theory, we remand to Part 1 of [1].
Remark 1.12.9. In the setting of Ω-Calculus, we can derive the axioms of Alpha-
Calculus:
• ACT0 and ACT1 are consequences of Axioms 1 and 2. In particular, real
sequences {xn}n∈N can be interpreted as functions f : N → R and, by extension
of functions, we can think of their Alpha-limit as the evaluation of f ∗ at some
infinite point in N ∗ . A good, somehow “natural” candidate is max(N ◦ ) that,
in our model, coincides with the number α = max(R ◦ ).
41

CHAPTER 1. Ω-CALCULUS
• It can be easily seen that, if we consider the identity function ι : N → N, then
the number ι ◦ (max(N ◦ )) is indeed a “new” number in the sense of ACT2.
• Finally, ACT3 holds because of Axiom 3.
We can conclude that Ω-Calculus is an “extension” of Alpha Theory: we have all
the tools from Alpha Calculus, plus some original notions and instruments.
42

Chapter 2
Selected topics of Ω-Calculus
Decisive progress in science often
depends on a change of direction.
Kurt Gödel
Since the main purpose of Ω-Calculus is to provide a theory strong enough
to allow the development of calculus, we find opportune to show how some
selected topics of calculus can be studied inside this theory. In the first
section of this chapter, we will see how the important notions of continuity and convergence
can be given by Ω-Calculus and, in the second section, we will characterize
the usual topology of the real line by the use of hyperfinite extensions of sets. This
characterization will then be used in the proof of some famous theorems of calculus
by the means of Ω-Calculus.
Our attention won’t be excessively focused on how “classical” mathematics can
be developed without the concept of classical limit and by using infinite and infinitesimal
numbers, since this is the realm of non standard analysis. Instead, our
goal is to show how some topics of “classical” mathematics can be interpreted by
Ω-Calculus, with a particular interest about how the use of Ω-limit and other instruments
presented in the previous Chapter can be used to prove some well known
theorems of calculus.
Of course, since there isn’t only one way to do mathematics, the reader could
wonder why do we think this approach is more interesting than the classical one.
The answer is rather simple: Ω-Calculus combines the intuitive aspects of nonstandard
analysis with other tools (namely, the Ω-limit and the hyperfinite extension
of sets) that, in our opinion, simplify a lot the complexity of the proofs. Clearly,
this simplification comes with a price: the necessity of at least basics knowledge of
Ω-Calculus. We will not discuss further wether the benefits exceed the downsides,
and will leave the reader with this open question.
In the course of this chapter, we will assume that the reader is familiar with
the standard wiewpoint of the topics treated. In particular, we will assume that
the reader knows the fundamental results of standard analysis regarding sequences,
series, limits, continuity and basic topology of the real line. We will often use [3] as
reference.
43

CHAPTER 2. SELECTED TOPICS OF Ω-CALCULUS
2.1 Convergence and continuity
Calculus has its limits.
Anonymous
The most powerful and characterizing tool of standard analysis is without any
doubt the notion of limit, and in particular that of limit of sequences and functions.
In this section, we will approach the idea of “classical limit” from the viewpoint
of Ω-Calculus. More precisely, we will study how the concept of convergence can
be developed with the instruments of Ω-Calculus. We begin with the study of
convergence in real sequences, and then we move on to the case of real, real valued
functions.
To fix the notation, we start with two definitions, that should be familiar to the
readers.
Definition 2.1.1. A real sequence is a function f : N → R. Sometimes we will
denote with {xi}i∈N the sequence f defined by f(i) = xi.
Definition 2.1.2. Let {xi}i∈N be a sequence and y ∈ R. The sequence converges to
y if
∀ɛ > 0 ∃m : ∀n ≥ m |xn − y| ≤ ɛ
In the context of Ω-Calculus, we want to introduce a new definition of convergence
consistent with Definition 2.1.2, that doesn’t use the idea of classical limit.
The basic idea is to check the behaviour of the sequence f “near infinity”. Since a
sequence is a function from N to R, we can consider its hyperfinite extension
f ◦ : N ◦ → R ∗
The intuitive idea of checking the behaviour of f near infiniy can be then translated
to evaluating the function f ◦ at some infinite hyperreal point. A somehow “natural”
candidate is the number max(N ◦ ) that, in our model, coincides with the number α,
and we explicitely observe that we will feel free to use this property in the next
definitions. Taking this into account, the first attempt to formalize the vague idea
of convergence expressed above gives rise to the following definition:
Definition 2.1.3. Let f be a sequence and ξ ∈ R ∗ a finite number. The sequence
converges to ξ if
f ◦ (α) ∼ ξ
The idea of this definition is to see the value that f ◦ assumes when calculated
in α, an infinite number. Of course, we need now to check if this definition is well
posed and if it is “a good definition of convergence”, in the sense that it has all the
properties we expect convergence to have. We begin to address this problem with
an example.
44

Example 2.1.4. Let’s consider the sequence
f(i) =
2.1. CONVERGENCE AND CONTINUITY
1 if i ∈ E
0 if i ∈ O
In the standard sense, this function doesn’t converge. In the sense of definition 2.1.3,
the function converges to 1 or to 0.
A definition of convergence so different than the standard one is not so useful: we
want the new definition to agree with the old one in deciding if a function converges
or doesn’t converge. Let’s try again: the idea is still the same, to check the behaviour
of f ◦ when calculated in an infinite number, but this time we want to avoid situations
like the one of example 2.1.4. To do so, we will check the behaviour of f ◦ at all
infinite points ζ ∈ N ◦ .
Definition 2.1.5. Let f be a sequence and ξ ∈ R ∗ a finite number. The sequence
converges to ξ if
f ◦ (ζ) ∼ ξ ∀ζ ∈ N ◦ : sh(ζ) = +∞
What happens if we consider again the sequence f of example 2.1.4? We can
find a function ϕ : Pfin(Ω) → N such as ϕ(λ) ∈ E for all λ ∈ Pfin(Ω) and, if
ξ = limλ↑Ω ϕ(λ), sh(ξ) = +∞. Analogously, we can find a function ψ : Pfin(Ω) → N
such as ψ(λ) ∈ O for all λ ∈ Pfin(Ω) and, if ζ = limλ↑Ω ψ(λ), sh(ζ) = +∞. From
the definition, we have: ξ and ζ ∈ N ◦ ,
and
f(ϕ(λ)) = 1 ∀λ ∈ Pfin(Ω)
f(ψ(λ)) = 0 ∀λ ∈ Pfin(Ω)
Now, Axiom 2 ensures that f(ξ) = 1 and f(ζ) = 0. According to definition 2.1.5,
f doesn’t converge neither to 0 nor to 1, so we can safely conclude that f doesn’t
converge. With a similar argument, we can show that if a sequence doesn’t converge
in the sense of definition 2.1.2, it doesn’t converge in the sense of definition 2.1.5.
Now, let’s check what happens to convergent sequences.
Example 2.1.6. Let’s consider the sequence f(i) = 1/i. We know that this sequence
converges to 0 in the standard sense. To see what happens with the new definition,
let’s take ϕ and ψ : Pfin(Ω) → N so that ξ = limλ↑Ω ϕ(λ), ζ = limλ↑Ω ψ(λ) and
sh(ξ) = sh(ζ) = +∞. From the definition of f, we have that both f(ξ) = 1/ξ and
f(ζ) = 1/ζ are infinitesimal numbers. Their difference is infinitesimal, too.
What can we conclude from the previous example? Simply: if f converges to ξ
according to Definition 2.1.5, then f converges to every ζ ∈ mon(ξ).
We now have all elements to formulate a definitive, “good” definition of convergent
sequence, that we will complete with the concepts of diverging sequence and
not converging sequence.
45

CHAPTER 2. SELECTED TOPICS OF Ω-CALCULUS
Definition 2.1.7. Let f be a sequence and y ∈ R. We will say that the sequence f
converges to y if
f ◦ (ζ) ∈ mon(y) ∀ζ ∈ N ◦ : sh(ζ) = +∞
We will say that f diverges positively if
sh (f ◦ (ζ)) = +∞ ∀ζ ∈ N ◦ : sh(ζ) = +∞
and we will say that f diverges negatively if −f diverges positively. If f behaves
differently, we will say that f does not converge.
Now, all the aberrant behaviours are out of the way, and we can work with this
final definition. We have chosen to show its genesis to give the reader an example of
the process behind the formation of a new mathematical concept: first, we started
with an intuitive, imperfect idea, which we improved during the development of the
theory. This process was cleverly noticed and clearly exposed for the first time by
Imre Lakatos in [16].
Before we move on, we want to remark that, thanks to Definition 2.1.7, if f
converges, there aren’t any doubts that f converges to a unique number: this will
be the only real number y satisfying the condition
f ◦ (ζ) ∈ mon(y) ∀ζ ∈ N ◦ : sh(ζ) = +∞
For the sake of completeness, we observe explicitely that, if there are two distinct
real numbers y1 and y2 such as
and
f ◦ (ζ) ∈ mon(y1) for some ζ ∈ N ◦ : sh(ζ) = +∞
f ◦ (ξ) ∈ mon(y2) for some other ξ ∈ N ◦ : sh(ξ) = +∞
then, by definition, f wouldn’t converge neither to y1 nor to y2. Moreover, if f and
g are sequences, Definition 2.1.7 and the axioms of Ω-Calculus ensure also that, if f
converges to y and g converges to z, then the sequence f + g converges to y + z, and
the same holds for the other algebraic operations. We recall that, in the standard
approach to convergence given by classical limit, those are theorems one needs to
prove.
In the next part of this section, we want to study the case of real, real valued
functions. First of all, we recall the standard definition of limit:
Definition 2.1.8. Let A ⊆ R, x0 an accumulation point for A and f : A → R. We
say that f converges to y as x approaches x0 if
∀ɛ > 0 ∃δ > 0 : ∀x ∈ A, x = x0, |x − x0| ≤ δ ⇒ |f(x) − y| ≤ ɛ
This definition sums up in a rigorous manner the informal idea of f(x) being
“very close” to y whenever x is “very close” to x0, without being equal to it.
Let’s pause for a moment and think about what we have written. We have
described informally the notion of convergence using the words “very close” but, in
our theory, we have a precise definition of “infinitely close” that express the same
idea formally. This allows us to translate the underlying concept of Definition 2.1.8
to the language of infinitesimal numbers and Ω-Calculus.
46

2.1. CONVERGENCE AND CONTINUITY
Definition 2.1.9. Let A ⊆ R, x0 an accumulation point for A and f : A → R. We
say that f converges to y as x approaches x0 if
∀ξ ∈ mon(x0) ∩ A ◦ , ξ = x0 ⇒ sh(f ◦ (ξ)) = y
We will say that f diverges positively as x approaches x0 if
∀ξ ∈ mon(x0) ∩ A ◦ , ξ = x0 ⇒ sh(f ◦ (ξ)) = +∞
and we will say that f diverges negatively as x approaches x0 if −f diverges positively.
If f behaves differently, we will say that f doesn’t converge as x that approaches x0.
We leave to the reader the easy task to extend Definition 2.1.9 to the cases where
x approaches ±∞.
It’s not immediate to see that the classical definition is equivalent to Definition
2.1.9, but it’s easy to recognize in the latter the intuitive idea of limit. Moreover,
this second definition seems easier to understand and remember than the classical
one, even by maths students. For further reference about this brief consideration, we
remand to [17], where the author studied the relationship between formal definitions
of mathematical concepts and the mental representation of those concepts by a
statistical population of undergraduate students in mathematics. In particular, some
of her research involved the mental representation of the concept of classical limit:
during some interviews with students who learned the definition of limit as stated in
Definition 2.1.8, she discovered that the formal definition was forgotten and replaced
with the intuitive idea “f has limit l as x approaches x0 if we get nearer and nearer
to l, but we never reach it”. In the words of some student, “When the variable,
the x approaches a certain point, the image of the function approaches a certain l”.
It seems to us that this intuitive idea of limit is closer to Definition 2.1.9 than to
Definition 2.1.8. This is the main reason of our interest in nonstandard presentations
of calculus: we are deeply fascinated by ways of do mathematics that combine the
formal precision of the discipline without putting to the test the basic intuitions that
seems rooted in the way humans think.
Back to the theory, we now want to show a useful result: to check if a function
converges when x approaches +∞, we can check the value of f in all infinite points
of R ∗ (and not only those in R ◦ ). Formally, we have:
Proposition 2.1.10. Let f : R → R, y ∈ R. f converges to y as x approaches +∞
if and only if for all ξ ∈ R ∗ : sh(ξ) = +∞, sh(f ∗ (ξ)) = y. The same holds if we
replace y with +∞ or −∞.
Proof. One of the implications is trivial. Let’s focus on the other.
Suppose that there is ξ ∈ R ∗ \ R ◦ with sh(ξ) = +∞ and f ∗ (ξ) ∼ y. If we find a
function ψ : Pfin(Ω) → R with ψ(λ) ∈ λ such as limλ↑Ω f(ψ(λ)) ∼ y, then the proof
would be concluded.
By hypothesis, ξ = limλ↑Ω ϕ(λ) for some ϕ : Pfin(Ω) → R. Without loss of
generality, we can also assume that, for some n ∈ N, |f(ϕ(λ)) − y| > 1/n is true for
all λ ∈ Pfin(Ω). If we define
A = {x ∈ R : x = ϕ(λ) for some λ ∈ Pfin(R)}
47

CHAPTER 2. SELECTED TOPICS OF Ω-CALCULUS
it’s clear that A contains arbitrarily large numbers: otherwise, we would have that
ξ < n for some n ∈ N, against the hypothesis. Moreover, by construction, we have
At this point, define
ψ(λ) =
f ∗ (A ∗ ) ∩ mon(y) = ∅ (2.1)
max(A ∩ λ) if A ∩ λ = ∅
max(λ \ {y}) otherwise
By definition, ψ(λ) ∈ λ for all λ ∈ Pfin(Ω), so that ζ = limλ↑Ω ψ(λ) ∈ R ◦ . It’s also
clear that ζ is an infinite number, thanks to the fact that A has no upper bound.
If we can prove that f ◦ (ζ) ∈ f ∗ (A ∗ ), then the thesis would follow by Equation
2.1. But this is easily shown: since the hypothesis λ ∩ A = ∅ is eventually true, we
deduce that f(ψ(λ)) ∈ f(A) is eventually true and, thanks to Proposition 1.7.2, we
obtain f ◦ (ζ) ∈ [f(A)] ∗ . We conclude by part 1 of Proposition 1.8.3.
In the case where f has limit ±∞, mutatis mutandis, the same proof can be
applied.
In the next example, we will see Definition 2.1.9 at work.
Example 2.1.11. Let’s consider the function f : (0, +∞) → R defined by f(x) =
sin(1/x). We want to calculate if f(x) converges for x that approaches 0. The idea
is quite easy: we know that sin(x) = 1 when x = 4k+1π,
k ∈ N, and sin(x) = −1
2
π, k ∈ N. Now, if we consider the two functions
when x = 4k+3
2
ϕ(λ) =
2
(4|λ| + 1)π
and ψ(λ) =
2
(4|λ| + 3)π
we have that limλ↑Ω ϕ(λ) ∼ limλ↑Ω ψ(λ) ∼ 0, but f(ϕ(λ)) = 1 and f(ψ(λ)) = −1 for
all λ ∈ Pfin(Ω). This implies
1 = lim
λ↑Ω f(ϕ(λ)) = lim
λ↑Ω f(ψ(λ)) = −1
so we can conclude that sin(1/x) doesn’t converges as x approaches 0.
Now, we will give the definitions of continuity and of uniform continuity. The
definitions are quite straightforward, and once again we will not prove that they are
equivalent to the standard ones. We can see that also those definitions are quite
similar to the respective intuitive ideas.
Definition 2.1.12. Let A ⊆ R, x0 ∈ A and f : A → R. We say that f is continuous
in x0 if
∀ξ ∈ A ◦ , ξ ∼ x0 ⇒ sh(f ◦ (ξ)) = y
We say that f is continuous in A if f is continuous at every point of A.
We say that f is uniformly continuous in A if f is continuous in A and
48
∀ξ, ζ ∈ A ◦ , ξ ∼ ζ ⇒ f ◦ (ξ) ∼ f ◦ (ζ)

2.2. TOPOLOGY BY Ω-CALCULUS
Now, we want to provide some examples of reasonings about uniform continuity.
Once again, we remark that the methods used appear to be fair simpler than the
standard ones.
Example 2.1.13. The same reasoning as in Example 2.1.11 shows that sin(1/x) is
not uniformly continuous.
Example 2.1.14. Let’s consider the function f : R → R defined by f(x) = x 2 . We
want to show that f is not uniformly continuous. Let’s consider the two numbers
α and α − 1/α: in our model, we know that both belong to R ◦ . By definition,
f ◦ (α) = α 2 , while f ◦ (α − 1/α) = α 2 − 2 + 1/α 2 . Their difference is 2 − 1/α 2 , and
this is a finite, non infinitesimal number, so we can conclude that f is not uniformly
continuous.
Now, consider f|[a,b), with a and b ∈ R. This time, let’s pick ξ ∈ [a, b) ◦ and ɛ
infinitesimal such that ξ − ɛ ∈ [a, b) ◦ . Let’s calculate the difference between their
images:
f(ξ) − f(ξ − ɛ) = ξ 2 − (ξ 2 − 2ξɛ + ɛ 2 ) = 2ξɛ − ɛ 2
Since ξ is finite, the last term of the equality chain is infinitesimal. We want to
explicitely remark that this result is independent of the choice of both ξ and ɛ, so we
obtained a (quite straightforward) proof that f|[a,b) is uniformly continuous.
Remark 2.1.15. The previous examples give us the chance to explicitely state the
following, crucial consideration:
Studying calculus with the tools of Ω-Calculus, we don’t need a “classical” idea of
limit. We can replace the operation of classical limit with evaluation of
hyperextended functions at some hyperreal points.
It’s important not to understimate this difference. We think that evaluate a
function at a point is way easier than fully master the “classical” definition of limit.
While our definition isn’t a new and more powerful tool than the “classical” one, it
has the potential to give easy access to the complex, yet intuitive idea behind the
concept of “classical” limit, without sacrificing formal rigor.
2.2 Topology by Ω-Calculus
Calculus required continuity, and
continuity was supposed to require
the infinitely little; but nobody
could discover what the infinitely
little might be.
Bertrand Russel
In the previous section, we have introduced the concept of continuity for functions.
The least requirement to define a notion of continuity that “makes some
sense” for real, real valued functions is a topology on R. A privileged choice of
49

CHAPTER 2. SELECTED TOPICS OF Ω-CALCULUS
topology on R which induces the known notion of continuity is the topology induced
by the distance d(x, y) = |x−y|, which we will denote (R, Td). With the instruments
of Ω-Calculus, however, we have based the notion of continuity upon the notion of
“two points being infinitely close”, as formally described in Definition 1.2.3. With
this idea in mind, we want to give R a topology that takes into account this notion.
To obtain this goal, we will follow [1] in endowing R with an uniform topology.
Before we proceed, we want to remark that the notion of uniform topology is
interesting on its own, because it can be generalized to arbitrary sets in which
we have defined somehow a notion of “infinite closeness”, and this topology alone
is sufficient to define a lot of interesting tool of calculus (e.g. various types of
convergence of function sequences) and to prove with agility some basic theorems
(e.g. Weierstrass theorem of maximum and minimum and Bolzano’s theorem of
zeros). For an introduction on this topic, we remand to [1].
We begin with the abstract definition of a uniform topology for R, and then
specialize it to our concrete case.
Definition 2.2.1. A Hausdorff uniform topology for R is an equivalence relation
∼τ over R ◦ , such that x ∼τ y for all x, y ∈ R. Given that, we define
and the family of open and closed sets:
monτ(x) = {ξ ∈ R ◦ : ξ ∼τ x}
• A ⊆ R is an open set if and only if x ∈ A implies monτ(x) ∈ A ◦ ;
• C ⊆ R is a closed set if and only if x ∈ C c implies monτ(x) ∩ A ◦ = ∅
It’s straightforward to verify that the two families defined satisfy all the usual
properties of open and closed sets, so that the definition is well posed.
In our case, a good candidate for ∼τ is the relation ∼ of being “infinitely close”,
introduced in Definition 1.2.3. Of course, we will need to restrict ∼, that is defined
on all R ∗ , only to the set R ◦ . With abuse of notation, we will often denote this
relation by ∼ and, in the same spirit, we will call mon(x) = {ξ ∈ R ◦ : ξ ∼ x}. When
the need of clarity arises, we will use the symbols ∼ ◦ and mon ◦ (x) instead.
With this choice for the relation ∼, we obtain the topology defined by
• A ⊆ R is an open set if and only if x ∈ A implies mon(x) ⊆ A ◦ ;
• C ⊆ R is a closed set if and only if x ∈ C c implies mon(x) ∩ A ◦ = ∅
We will denote this topology by (R, T ).
Now, we will show that this topology is equivalent to (R, Td).
Proposition 2.2.2. A ⊆ R is open (respectively, closed) in the topology (R, T ) if
and only if A is open (respectively, closed) in the topology (R, Td).
Proof. Let’s suppose that A is open in (R, T ): then, for all x ∈ A, mon(x) ⊆ A ◦ .
We want to show that there exists some r ∈ R such that the set Br(x) = {y ∈ R :
|x − y| < r} ⊆ A. Let’s suppose that this is not the case, so that, for all r ∈ R,
Br(x) ⊆ A. This implies that there exists r0 ∈ R and a point z ∈ R such that
50

2.2. TOPOLOGY BY Ω-CALCULUS
z ∈ Br \ A for all r > r0. Without loss of generality, we can suppose z > x, so we
can deduce A ∩ (x, z] = ∅. This implies that, for instance, x + 1/α ∈ A ◦ but, since
x + 1/α ∈ mon(x), this is a contradiction.
On the other hand, if A is open in (R, Td), then, if we choose x ∈ A, there exists
r0 such that Br(x) ⊆ A for all r < r0. For every ξ ∈ mon(x), there exists a function
ϕξ such that limλ↑Ω ϕξ(λ) = ξ, and this hypothesis ensures that ϕξ(λ) ∈ Br(x) is
eventually true for all r ∈ R. Considering only r < r0, we get that ϕξ(λ) ∈ A is
eventually true and, by the arbitrariety of ξ, this implies mon(x) ⊆ A ◦ .
If A is a closed set, then use the fact that A c is an open set and the previous
part of the Proposition.
With this result at hand, it’s natural to wonder why have we chosed to introduce
the usual topology (R, Td) in such a convoluted way. The answer will be clear after
we’ve seen the proofs of some classical results stated in the language of this topology.
Before we do so, we proceed by introducing some other topological concepts.
Now we will define compact sets. Instead of giving the usual characterization
involving finite coverings, we take another route: we begin by introducing the idea
of a near-standard point for a set, and then we will define compact sets using nearstandard
points.
Definition 2.2.3. We say that ξ ∈ A ◦ is near-standard in A if sh(ξ) ∈ A.
Definition 2.2.4. We say that K ⊆ R is compact if and only if every ξ ∈ K ◦ is
near standard in K.
Using the characterization of compacts with respect to the topology induced by
the metric in R (namely, K ⊆ R is compact if and only if K is closed and bounded),
it’s easy to see that K is compact in the usual sense if and only if K is compact
according to Definition 2.2.4.
The concept of near-standard points is also useful in characterizing connected
sets.
Definition 2.2.5. We say that C ⊆ R is connected if and only if, for every decomposition
{C1, C2} of C, there exists ξ1 ∈ (C1) ◦ and ξ2 ∈ (C2) ◦ , both near standard in
C, such that ξ1 ∼ ξ2.
In order to conclude the premises of this section, we will give the definitions of
accumulation point for a set, internal point and frontier point.
Definition 2.2.6. Let A ⊆ R and x ∈ A. We say that
• x is an accumulation point for A if {x} ⊂ mon(x) ∩ A ◦
• x is an internal point of A if mon(x) ⊆ A ◦
• x is a frontier point for A if mon(x) ⊂ A but mon(x) ∩ A = ∅
It’s easily seen that all definitions given agree with the usual ones, so we won’t
give the easy proofs of the equivalence. Instead, we move on and show how these
new definitions allow to prove some classical results with ease.
51

CHAPTER 2. SELECTED TOPICS OF Ω-CALCULUS
Theorem 2.2.7 (Weierstrass Theorem). If K ⊂ R is a nonempty compact set and
f : K → R is continuous, then f has maximum and minimum.
Proof. For all λ ∈ Pfin(Ω), if λ ∩ K = ∅, f|λ has maximum and minimum. Since
this hypothesis is eventually true, let xλ ∈ λ such that f(xλ) is a maximum of
f|λ. Now, let’s consider ξ = limλ↑Ω xλ. By definition, ξ ∈ K ◦ and it’s easy to see
that f ◦ (ξ) ≥ f(z) for all z ∈ K. Since K is compact, ξ is near standard in K, so
sh(ξ) = x ∈ K. The continuity of f ensures that f(x) ∼ f ◦ (ξ), and from this we
conclude f(x) ≥ f(z) for all z ∈ K.
The existence of a minimum follows by the fact that −f has a maximum.
Theorem 2.2.8 (Bolzano). If C ⊆ R is connected and f : C → R is a continuous
function, then f(C) is connected.
Proof. Suppose that f(C) = C1 ∪ C2: we want to show that there exist two nearstandard
points ζ1 ∈ (C1) ◦ and ζ2 ∈ (C2) ◦ such that ζ1 ∼ ζ2. If C1 ∩C2 = ∅, then the
thesis is trivial. Let’s suppose C1 ∩C2 = ∅: clearly, this hypothesis implies f −1 (C1)∩
f −1 (C2) = ∅. By Corollary 1.7.10, we have also that [f −1 (C1)] ◦ ∩ [f −1 (C2)] ◦ = ∅.
Since C is connected, there exists ξ1 ∈ [f −1 (C1)] ◦ and ξ2 ∈ [f −1 (C2)] ◦ and x ∈ C
such that ξ1 ∼ x ∼ ξ2. Without loss of generality, we can assume f(x) ∈ C1, so we
can choose ζ1 = f(x). If mon(f(x)) ∩ (C2) ◦ = ∅, we can conclude by choosing ζ2 in
that intersection. On the other hand, the hypothesis mon(f(x)) ∩ (C2) ◦ = ∅ implies
that f(x) is an internal point of C1, and so, because f is continuous, x is an internal
point of f −1 (C1). This would imply that ξ2 ∈ [f −1 (C1)] ◦ , but this in contradiction
with the fact that f −1 (C1) and f −1 (C2) are disjoint.
Corollary 2.2.9 (Bolzano Theorem of Zeroes). Let f : [a, b] → R such that f(a) ·
f(b) < 0. Then, there exists c ∈ [a, b] such that f(c) = 0.
Proof. Without loss of generality, let’s suppose that f(a) < 0 < f(b). By Theorem
2.2.8, f([a, b]) is connected, and this implies that [f(a), f(b)] ⊆ f([a, b]). In
particular, 0 ∈ f([a, b]).
Theorem 2.2.10 (Heine). If K ⊂ R is a compact set and f : K → R is continuous,
then f is uniformly continuous.
Proof. Let ξ and ζ ∈ K ◦ with ξ ∼ ζ. Since K is compact, there exists x ∈ K
verifying ξ ∼ x ∼ ζ. Then, by the continuity of f at x, we deduce f(ξ) ∼ f(x) ∼
f(ζ).
We encourage the reader to compare the proof of this proposition with some
standard ones (for an example, see Theorem 6.3 of [3]).
Now, we want to prove Banach-Caccioppoli Fixed Point Theorem. We recall
that f : R → R is a contraction if there exists 0 < k < 1 such that, for any x,
y ∈ R, |f(x) − f(y)| ≤ k|x − y|; and that x0 ∈ R is a fixed point for f if f(x0) = x0.
Observe that, by definition, a contraction is continuous.
Theorem 2.2.11 (Banach-Caccioppoli Fixed Point Theorem). Every contraction
f : R → R has a unique fixed point.
52

2.2. TOPOLOGY BY Ω-CALCULUS
Proof. The unicity of the fixed point is clear, so we will only prove its existence.
Let’s pick x ∈ R and define ξ = limλ↑Ω f |λ| (x) (f |λ| is the |λ|-th iteration of f).
Clearly, ξ ∈ R ∗ . By hypothesis,
|f |λ|+1 (x) − f |λ| (x)| ≤ k|f |λ| (x) − f |λ|−1 (x)|
is true for all λ ∈ Pfin(Ω) and, iterating the inequality, we get
|f |λ|+1 (x) − f |λ| (x)| ≤ k |λ| |x − f(x)|
Since f ∗ (ξ) = limλ↑Ω f |λ|+1 (x), this result implies that f ∗ (ξ) ∼ ξ. If we show that ξ
is finite, then, using the continuity of f and Proposition 2.1.10, from f ∗ (ξ) ∼ ξ we
deduce that sh(ξ) is the fixed point for f.
Now, we will prove the finiteness of ξ by reductio ad absurdum. Suppose that ξ
is infinite: then, for any x ∈ R, we would have
|ξ + ɛ − f(x)| = |f(ξ) − f(x)| ≤ k|ξ − x|
where ɛ is an infinitesimal number. Since ξ is infinite, clearly ξ > (ɛ − f(x)) and
ξ > x. From those inequalities, we deduce
that immediately leads to
ξ + ɛ − f(x) ≤ k(ξ − x)
(1 − k)ξ ≤ kx + f(x) − ɛ
and this is a contradiction, since the first member of the inequality is an infinite
number and the second is finite. We conclude that ξ is finite, as we wanted.
We want to explicitely observe some differences between this proof and the classical
one. In the classical proof, one picks an x ∈ R and defines the sequence
{f n (x)}n∈N; then, the proof revolves around showing that this is a Cauchy sequence
and therefore has a finite limit. In our proof, the existence of ξ is guaranteed by
the axioms of the theory, and the only thing we need to show is that ξ is finite. It
turns out that showing that ξ is finite involves only basic algebraics operations and
a little familiarity with the concept of finite and infinite numbers.
Overall, we believe that the proofs of Banach-Caccioppoli Fixed Point Theorem
and many other famous theorems are simplified in the mathematical framework of
Ω-Calculus. It could be interesting to inquire if this happens only in a few cases or
if it’s a general feature of Ω-Calculus. In the second case, then, it would arise the
question if Ω-Calculus could be used as a didactic tool to help students understand
mathematics. We hope that, once one gets confidence with the operation of Ω-limit,
then the development of the theory could follow an easier route. It’s still an open
question, and a very interesting one, to find out if this is the case.
53

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
Chapter 3
Integrals by Ω-Calculus
Conjectures and concepts both
have to pass through the purgatory
of proofs and refutations. Naive
conjectures and naive concepts are
superseded by improved conjectures
and concepts growing out of the
method of proofs and refutations.
Imre Lakatos
Historically, the notion of integral is deeply connected with the attempts to
calculate the area underneath curves. One of the first techniques developed
for this goal was Cavalieri’s method of the “indivisibles”, where the
Italian mathematician assumed that an area was composed by an infinite number of
lines with no width. In the practice, Cavalieri’s method consisted in approximating
the area underneath a curve with the area of a certain number of rectangles and
then evaluating the area of those rectangles as their number increases. As we can
see, the underlying ideas of Cavalieri’s method somehow survived in the definition
of Riemann integral, where the “area underneath a curve”, when well defined, is
given as the infimum of the area of finite unions of rectangles that approximate the
curve from above or as the supremum of the area of finite unions of rectangles that
approximate the curve from below.
In our approach to integrals, we think that the original idea expressed by Cavalieri
can be formalized by Ω-Calculus: if we can give a meaningful definition of “area of
a line” then, by a process of “summing all the lines”, we hope to get an interesting
definition of integral.
To do so, at first we find opportune to define a concept of infinite sums that
have meaning for all subsets of R: in the first section of this chapter, we will briefly
present this idea, that will be a generalization by Ω-limit of the concept finite sums.
Then, we will define the “Ω-integral” by formalizing the intuitive ideas expressed
above, and subsequently we will show some of its properties. We will see that, thanks
to Ω-limit, the Ω-integral will have many of the good properties we expect integrals
to satisfy.
At this point, a very natural question arises: has the Ω-integral some meaningful
54

3.1. HYPERFINITE SUMS
relations with the “classical” integrals over R, namely the Riemann integral and
Lebesgue integral? It turns out that, when the Riemann integral of a function is
well defined, then the Ω-integral is infinitely close to it. In the last section, we
will try to estabilish some results towards the proof of a similar relation between
Ω-integral and Lebesgue integral. Unfortunately, we will encounter some difficulties,
so we will leave the open question if Numerosity Axiom 1 is sufficient to entail such
a result.
3.1 Hyperfinite sums
In order to define the Ω-integral, we want to give a meaningful definition of “infinite
sums” over arbitrary subsets of R. The natural idea is to try to extend the concept
of finite sums by using Ω-limit. Following this intuition, we define:
Definition 3.1.1. For all E ⊆ R and for all f : E → R, we set
f(x)
x∈E ◦
f ◦ (x) = lim
λ↑E
and we will call it the hyperfinite sum over E◦ . We warn the reader that, with abuse
of notation, we will often write
f(x)
instead of the correct
x∈E ◦
f ◦ (x)
x∈E ◦
It naturally arises the question wether, if we choose E = N, there are some
differences between the hyperfinite sum over N◦ and the usual concept of serie.
We will address this question with an example: we know that, with the classical
definition of serie, the writing
k∈N (−1)k is meaningless, but, according to definition
3.1.1,
k∈N◦(−1) k is well defined and, in our model, we can even calculate it.
Example 3.1.2. We want to calculate
x∈N◦(−1) k . In our model, we have that
(−1) k
= lim (−1) k
⎛
= lim ⎝
(−1) k
⎞
⎠
k∈N ◦
But we know that
λ↑N
k∈λ
k∈λn,θ∩N
(−1) k =
x∈λ
λn,θ↑N
n
(−1) k
k=1
k∈λn,θ
and, since n = m! for some m ∈ N, n is even whenever n = 1. This ensures that
(−1) k = 0
k∈λn,θ∩N
55

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
is eventually true. We can apply transfer principle and conclude that, in our model
(−1) k = 0
x∈N ◦
We will not study in further details this concept of serie. Instead, we will only
state that, if
x∈E f(x) is well defined and absolutely convergent in the standard
sense, then
sh f(x) =
f(x)
x∈E ◦
and the proof of this statement is left as an exercise for the interested reader. The
request that
x∈E f(x) must be absolutely convergent is due to the fact that the
definition of
x∈E◦ f(x) involves reordering the terms of the original sum, and we
know from basic calculus that, if the serie is not absolutely convergent, then its value
is not independent from the order of its terms.
3.2 Ω-integral
As we have anticipated in the introduction of this chapter, we would like to give
a definition of integral of a function f over a set E ⊆ R as an infinite sum of the
“areas” of the lines with base all the singletons {x} ⊆ E and height f(x). In the
previous Section, we have defined the kind of infinite sums we’d like to use, namely
the hyperfinite sums. The only ingredient we need to define is this “area”.
For a matter of commodity, suppose that E = [a, b) and that the function f
is the constant c1. In this case, the integral of c1 over [a, b) assumes the meaning
of a “measure” for the set [a, b), and our wish is that this “measure” is at least
infinitesimally close to the expected value b − a. Thanks to Numerosity Axiom 1,
to ensure the validity of this result, it’s sufficient that the integral of the constant
function c1 over a set [a, b) is equal to n([a, b)) · n([0, 1)) −1 . To ensure that this
equality holds, we can define the integral in the following way:
x∈E
Definition 3.2.1. Let f : E ⊆ R → R. We define
f ◦ 1
(x)dΩx =
f
n([0, 1))
◦ (x)
E ◦
This definition is well posed for all functions, because
x∈E◦ f ◦ (x) is well defined
for all functions and for all subsets of R. In the future, with abuse of notation, we
will often write
f(x)dΩx
instead of the correct
E ◦
E ◦
f ◦ (x)dΩx
Now, we need to check if this integral shares at least some of the good properties
of the Riemann and Lebesgue integrals. We will start the study of the properties of
56
x∈E ◦

the Ω-integral by showing that
[a,b) ◦
c1(x)dΩx =
n([a, b))
n([0, 1))
More precisely, we will show that, for all E ⊆ R,
c1(x)dΩx = n(E)
n([0, 1))
since this is sufficient to entail the validity of Equation 3.1.
Proposition 3.2.2. For all E ⊆ R,
c1(x)dΩx = n(E)
n([0, 1))
E ◦
E ◦
Proof. By definition, we have that
E ◦
c1(x)dΩx =
1
n([0, 1)) lim
c1(x)
λ↑E
x∈λ
3.2. Ω-INTEGRAL
Since λ is finite, it’s straightforward to see that the following equality holds:
c1(x) = |λ|
x∈λ
Taking the Ω-limit and multypling by n([0, 1)) −1 , we obtain the thesis.
(3.1)
Since, thanks to the relation given by Proposition 3.2.2, we will make a lot of
use of the ratio n(E) · n([0, 1)) −1 , for a matter of commodity we will denote it by a
specific symbol.
Definition 3.2.3. If E ⊆ R, we will call the normalized numerosity of order 1 of
the set E the ratio
n(E)
n([0, 1))
and we will denote it by nn1(E).
We find useful to show a simple example of the use of this new notation.
Example 3.2.4. With the notation of Definition 7.2.2, Numerosity Axiom 1 says
that, for all a, b ∈ Q, nn1([a, b)) = b − a, and Proposition 3.2.2 says that, for all
E ⊆ R,
E ◦
c1(x)dΩx = nn1(E)
Now that we have defined the basic concepts, the remaining part of this section
is dedicated to the study some basic properties of Ω-integral. From now on, unless
we say otherwise, E will be an arbitrary subset of R. First of all, we check that the
Ω-integral is linear.
57

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
Proposition 3.2.5. For all f, g : E → R and for all a, b ∈ R,
(af + bg)(x)dΩx = a · f(x)dΩx + b · g(x)dΩx
E ◦
Proof. By definition, we have
(af + bg)(x)dΩx =
E ◦
and, by Corollary 1.8.4,
1
n([0, 1))
(af + bg) ◦ (x) =
x∈E ◦
E ◦
1
n([0, 1))
1
n([0, 1))
E ◦
(af + bg) ◦ (x)
x∈E ◦
x∈E ◦
(af ◦ (x) + bg ◦ (x))
From now on, we will omit the term n([0, 1)) −1 . Again by definition, we have
(af ◦ (x) + bg ◦
(x)) = lim (af(x) + bg(x))
x∈λ
x∈E ◦
but, since the sum is finite, we have
lim (af(x) + bg(x)) = lim a ·
λ↑E
λ↑E
f(x) + b ·
g(x)
and, using Ω-Axiom 3 several times, we get
lim a ·
λ↑E
f(x) + b ·
g(x) = a · lim f(x)
λ↑E
x∈λ
x∈λ
λ↑E
x∈λ
x∈λ
x∈λ
x∈λ
+ b · lim
λ↑E
g(x)
Now, multiplying the last member of the equality chain by n([0, 1)) −1 , we obtain, by
definition
a · f(x)dΩx + b · g(x)dΩx
as we wanted.
E ◦
As expected, the Ω-integral of a function over the union of disjoint sets is equal
to the sum of the integrals of the function over those sets. Formally:
Proposition 3.2.6. For all f : E → R and for all E1, E2 ⊆ E such as E1 ∩ E2 = ∅,
f(x)dΩx = f(x)dΩx + f(x)dΩx
E1∪E2
Proof. Since the relation
x∈λ∩(E1∪E2)
E1
f(x) =
x∈λ∩E1
E ◦
E2
f(x) +
x∈λ∩E2
f(x)
holds for all λ ∈ Pfin(Ω), we can apply Transfer Principle and conclude.
58
x∈λ

3.3. Ω-INTEGRAL AND RIEMANN INTEGRAL
As a consequence, we deduce that the same result holds for a finite union of
disjoint sets.
Corollary 3.2.7. For all f : E → R and for all finite families {Ei}i∈{1,...,n} such as
Ei ⊆ E for all i ∈ {1, . . . , n} and Ei ∩ Ej = ∅ whenever i = j,
n
i=1 Ei
f(x)dΩx =
n
i=1
Ei
f(x)dΩx
The result of Corollary 3.2.7 shouldn’t come as a surprise, since it’s very similar
to the result of finite additivity of numerosity stated by Corollary 1.6.3.
3.3 Ω-integral and Riemann integral
The definition of the Ω-integral reflects an intuitive, yet naive idea of area behind a
curve. It’s only natural to wonder if this integral has some mathematical interest and
has some relations with Riemann and Lebesgue integrals. In this section, we want
to show that there is indeed a close similarity between the Ω-integral and Riemann
integral. In order to distinguish the different integrals, we will denote Riemann
integral over a set E with the symbol R
, and Lebesgue integral with E E .
Since the notion of a function being Riemann-integrable is given considering step
functions, we begin by recalling that definition.
Definition 3.3.1. A function s : [a, b) ⊂ R → R is a step function if there are
x0, x1, . . . , xn ∈ [a, b] satisfying x0 = a, xn = b and xi < xj whenever i < j and if
there are y1, . . . , yn ∈ R such as the following equality holds:
s(x) = yi whenever x ∈ [xi−1, xi)
Thanks to Proposition 1.6.7, it’s easy to see that the Ω-integral of step functions
is infinitesimally close to their Lebesgue integral.
Proposition 3.3.2. Let s : [a, b) ⊂ R → R be a step function. Then
s(x)dΩx ∼ s(x)dx
[a,b) ◦
Proof. Since the domain is bounded and s assumes only a finite number of values
over [a, b), by linearity we can assume that s is constant over [a, b). Then, there
exists y ∈ R such as
s(x)dΩx = y · c1(x)dΩx = y · c1(x)dΩx
[a,b) ◦
[a,b) ◦
[a,b)
is true. By Proposition 3.2.2, we deduce that
y · c1(x)dΩx = y · nn1([a, b))
[a,b) ◦
[a,b) ◦
59

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
and, using Proposition 1.6.7, we can conclude
y · nn1([a, b)) ∼ y (b − a) =
as we wanted.
[a,b)
s(x)dx
We recall that a function f is Riemann-integrable over the set [a, b) if there
exists a real number l with the following property: for every positive ɛ ∈ R there
exists a positive δ ∈ R such that for every partition P = a = x0 < . . . < xs = b with
defined by
maxi=1,...,s(|xi − xi−1|) < δ, the step functions s ±
P
and
verify R
s +
P (x) = max
x∈[xi,xi+1) f(x) if x ∈ [xi, xi+1)
s −
P (x) = min f(x) if x ∈ [xi, xi+1)
x∈[xi,xi+1)
[a,b)
s ±
P
(x)dx − l
< ɛ
If such a number exists, we say that l is the integral of f over the set [a, b), and we
write
l =
R
[a,b)
f(x)dx
Proposition 3.3.3. If f : [a, b) → R is Riemann-integrable, then
R
f(x)dΩx ∼ f(x)dx
[a,b) ◦
Proof. If we pick k ∈ N, then there exists λk ∈ Pfin(Ω) such as the partition
Pλk = λk ∩ [a, b) verifies
R
(x)dx − l
< 1/k
[a,b)
[a,b)
s ±
Pλ k
and the formula is still true if we replace λk with another λ ⊇ λk. Now, Proposition
3.3.2 ensures that, for all λ ∈ Pfin(Ω), the following equality holds:
R
s ±
(x)dx = sh
Pλ s ±
Pλ (x)dΩx
and, in our case, this means that
sh
[a,b) ◦
[a,b)
[a,b) ◦
s ±
(x)dΩx
Pλk
− l
< 1/k
is eventually true for all k ∈ N. Applying Transfer principle, we can conclude
lim sh
λ↑Ω
s ±
(x)dΩx
Pλk ∼ l
60
[a,b) ◦

[a,b) ◦
3.4. NUMEROSITY AND LEBESGUE MEASURE
Another application of Transfer principle ensures that
lim sh
λ↑Ω
s −
(x)dΩx
Pλk ≤ f(x)dx ≤ lim sh
λ↑Ω
Combining the two results, we get
as we wanted.
[a,b) ◦
[a,b) ◦
f(x)dΩx ∼ l
[a,b) ◦
3.4 Numerosity and Lebesgue measure
s +
(x)dΩx
Pλk The result that allowed to relate Ω-integral to Riemann integral was given by Proposition
3.3.2, whose validity depends upon Proposition 3.2.2. We recall that this
Proposition states the equality between the Ω-integral of the characteristic function
of a set with the normalized numerosity of that set: in the case of intervals, Numerosity
Axiom 1 ensures that this value is the Peano-Jordan measure of the interval. For
this reason, we find interesting to begin our study of the relation between Ω-integral
and Lebesgue integral by studying the relations between numerosity and Lebesgue
measure. In particular, we begin by addressing a specific problem of numerosity:
given Numerosity Axiom 1, it’s hard to determine if numerosity is countably additive
or even countably subadditive. Indeed, it’s easier to show that the other inequality
is more likely to hold.
Example 3.4.1. Suppose that {En}n∈N are nonempty subsets of R and that Ei∩Ej =
∅ whenever i = j. If we call E =
n∈N En, then, by the trivial inclusion
we get that, for all k ∈ N,
k
En ⊂ E
n=1
k
n(En) < n(E)
n=1
If we have the additional hypothesis that nn1(En) ∈ R for all n ∈ N, we can go
further: by transfer of inequalities, we deduce
nn1(En) ≤ sh (nn1(E))
n∈N ◦
The best we can do is trying to get an additivity result under some hypotheses
of finiteness. We summarize those hypotheses in the next definitions.
Definition 3.4.2. We will call E + the family of all sets E =
n∈N [an, bn) satisfying
the hypothesis
1. E ⊆ [−N, N) for some N ∈ N;
61

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
2. {[an, bn)}n∈N are pairwise disjoint;
3. bn ≤ an+1 for all n ∈ N.
and we will call E − the family of all sets E =
n∈N [an, bn) satisfying hypothesis 1, 2
and
3’ bn+1 ≤ an for all n ∈ N.
Notice that, for all E ∈ E ± , the serie
n∈N (bn − an) is well defined in the
classical sense and assumes a finite value. Notice also that
n∈N (bn − an) is exactly
the Lebesgue measure of E. For a matter of convenience, we will give a name to
this measure.
Definition 3.4.3. We will denote by µ the Lebesgue measure over R. We remark
explicitely that, for all bounded E =
n∈N [an, bn), where the union is disjoint,
µ(E) =
n∈N (bn − an).
The idea is to give a result of countable additivity with respect to all sets E ∈ E ± .
There is only one problem: under those hypothesis, nn1([an, bn)) ∈ R ∗ , so we won’t
be able to calculate their hyperfinite sum. The solution is to consider only the sets
E ∈ E ± for which nn1([an, bn)) ∈ R for all n ∈ N: thanks to Numerosity Axiom 1,
we know that this request means that an and bn must be rational numbers.
Definition 3.4.4. We will call E ±
Q the family of all sets E ∈ E ± satisfying the
additional hypothesis an, bn ∈ Q for all n ∈ N.
We can now give a result of countable additivity for all E ∈ E ±
Q . Then, we will
be able to generalize it to all sets of E ± thanks to monotony of numerosity.
Proposition 3.4.5. For all E ∈ E +
Q , it holds
sh (nn1(E)) ∼
(bn − an)
n∈N ◦
Proof. Observe that, by hypothesis, for the classical limits it holds
lim
n→∞ an = lim bn
n→∞
Now, define b = limn→∞ bn and, for all k ∈ N, choose Ak and Bk ∈ Q such that
and
0 ≤ ak+1 − Ak < 1/k
0 ≤ Bk − b < 1/k
Explicitely, Ak and Bk are two rational points that satisfy
and
62
lim
n→∞ Ak = lim Bk = b
n→∞
[Ak, Bk) ⊇ [ak+1, b) ⊇
[an, bn)
n>k

3.4. NUMEROSITY AND LEBESGUE MEASURE
From the inclusion above, we deduce
[an, bn) ⊆ E ⊆ [an, bn) ∪ [Ak, Bk)
n≤k
hence, by monotony and finite additivity of numerosity,
(bn − an) ≤ nn1(E) ≤
(bn − an) + (Bk − Ak)
n≤k
n≤k
Notice that, by hypothesis, the first and the last term of the chain of inequalities
are rational numbers. We can then take the shadow of all the terms and obtain:
(bn − an) ≤ sh(nn1(E)) ≤
(bn − an) + (Bk − Ak)
n≤k
At this point, taking the Ω-limit of the previous equation with respect to k, we
obtain
(bn − an) ≤ sh(nn1(E)) ≤
(bn − an) − lim B|λ| − A|λ|
n∈N ◦
n≤k
n∈N ◦
If we prove that limλ↑Ω B|λ| − A|λ| ∼ 0, then the thesis will follow.
Suppose that limλ↑Ω B|λ| − A|λ| ∼ l: by the equivalence with classical limit, the
following equality holds in the standard sense:
n≤k
λ↑Ω
l = lim
n→∞ (Bn − An) = lim
n→∞ Bn − lim
n→∞ An = b − b = 0
We deduce l = 0, and this concludes the proof.
Corollary 3.4.6. For all E ∈ E +
Q , sh (nn1(E)) = µ(E).
Proof. This result follows by Proposition 3.4.5 and the relation
(bn − an) ∼ µ(E)
n∈N ◦
Putting everything together, we obtain
sh (nn1(E)) ∼ µ(E)
but, since both are real numbers, they must be equal.
With the next proposition, we will extend this result to all the sets in E + .
Proposition 3.4.7. For all E ∈ E + , sh (nn1(E)) = µ(E).
Proof. Thanks to Corollary 3.4.6, we only need to prove the thesis for all the sets
E ∈ E + \ E +
Q . To do so, choose such a set E and, notice that for all n ∈ N, we can
easily find two sets E + n and E − n ∈ E +
Q satisfying
E − n ⊆ E ⊆ E + n
63

CHAPTER 3. INTEGRALS BY Ω-CALCULUS
and
|µ(E ± n ) − µ(E)| < 1
(3.2)
n
By the first chain of inclusions and by monotony of numerosity, we deduce that the
following inequality
holds for all n ∈ N or, in other words,
sh(nn1(E − n )) ≤ sh(nn1(E)) ≤ sh(nn1(E + n ))
µ(E − n ) ≤ sh(nn1(E)) ≤ µ(E + n )
Taking the Ω-limit with respect to n and thanks to equation 3.2, we conclude
sh(nn1(E)) ∼ µ(E)
Now, notice that both numbers are real numbers, hence the desired equality.
Mutatis mutandis, the same arguments can be applied to prove the analog of
Proposition 3.4.5 and Corollary 3.4.6 where E ∈ E −
Q , and of Proposition 3.4.7 for
the case where E ∈ E − . For this reason, we will omit the proof of those statements,
that are left as exercises for the interested reader. Instead, we jump directly to the
final result, obtained from the previous ones by finite additivity of numerosity.
Proposition 3.4.8. If E = k
n=1 Ek and En ∈ E + ∪ E − for all n ≤ k, then
sh(nn1(E)) =
k
µ(Ek)
A few words of comment are in order. The final result stated in Proposition 3.4.8
is very narrow, and it’s very hard to use it to link numerosity to Lebesgue measure.
The main difficulty is that Lebesgue measure is obtained by considering arbitrary
unions of intervals, whereas in our case we can show how numerosity behaves only in
a very specific case. The restrictive hypotheses on the union considered don’t allow
an easy extension of this result to arbitrary unions, and it’s still an open question if
Numerosity Axiom 1 is sufficient to estabilish a meaningful relation between numerosity
and Lebesgue measure. Given the difficulties, we choose to leave this question
open for further research. The interested reader could find some more ideas toward
this direction in Sections 7.3 and 7.8 of this paper.
64
n=1

Chapter 4
Ω-Theory
A mathematician, like a painter or
a poet, is a maker of patterns.
G. H. Hardy
As we’ve seen in the previous chapters of this paper, the scope of Ω-Calculus
is limited to real functions, subsets of real numbers and their hyperextensions,
so that theory alone is insufficient for many applications. To
solve this issue, we can generalize the axiomatics of Ω-Calculus by simply extending
the range of the postulated properties from functions with range in the real numbers
to arbitrary functions. Clearly, we will require that, when we focus on real valued
functions, the Ω-limit will coincide to the one we studied in the previous chapters.
We will obtain a new theory, that we will call Ω-Theory, that extends Ω-Calculus
and gives more mathematical tools that can be applied in many different situations.
The only difference with Ω-Calculus will be that, in the basis of Ω-Theory, we
will omit the numerosity axioms and we will rule only those regarding Ω-limit. This
choice is due to the fact that different applications could require different axioms for
numerosity, and we want to be free to choose them as the need arises. Moreover, it
seems to us that numerosity axioms are more arbitrary and less “intrinsic” to the
theory as those of Ω-limit.
After a brief section about the foundational framework of Ω-Theory, we will follow
the same structure as Chapter 1: at first, we will present the axioms of the theory,
and then we will study their consequences, generalizing concepts already introduced
in the previous chapters and exploring new ideas. At last, we will exhibit a model
for Ω-Theory.
4.1 Foundational framework
The axioms of Ω-Calculus will have a really broad range, in the sense that they will
rule the Ω-limits of all sequences of mathematical objects. Here, by “mathematical
objects” we mean all the entities that are used in the ordinary practice of mathematics,
namely numbers, sets, functions, relations, ordered tuples, and so forth.
As suggested by the common use, in our approach we distinguish between sets as
65

CHAPTER 4. Ω-THEORY
legitimate collections of objects, and atoms as mathematical objects that are not
sets. We remark that, in the practice, the “status” of atom or set may depend on
the context. For instance, in the well-known construction of the real numbers as
Dedekind cuts, real numbers are defined as suitable subsets of rational numbers. On
the other hand, in the practice of analysis, one hardly considers a real number as a
set. Similarly, when studying a topological space, one takes its elements as atoms,
even in the cases when they are in fact sets (e.g. in the case of space of functions).
As a general rule, we shall make the assumption that numbers are atoms, and
that ordered tuples, relations and functions are sets. In fact, each of these notions
admits a representation as a set, and can be formally defined within the framework
of axiomatic set theory. In particular, we agree on the following:
• Ordered pairs are given by the Kuratowski pairs:
(a, b) = {{a}, {a, b}}
Inductively, ordered n + 1-tuples are defined by putting
(a1, . . . an+1) = ((a1, . . . , an), an+1)
• A binary relation R is identified with the set of ordered pairs that satisfy it:
R = {(a, b) : aRb}
In the same spirit, a n-place relation is the set of ordered n-tuples that satisfy
it.
• A function f is identified with its graph:
f = {(a, b) : b = f(a)}
At this stage, it is safe to say that when applying the Ω-Theory in everyday
practice, one does not need to know about the specific foundational framework that
is adopted. However, we signal that ZFC set theory is not the appropriate framework
for Ω-Theory: in particular, the axiom of foundation is incompatible with Ω-Theory,
and in Section 4.4 we will give an intuitive idea of why it is so. A convenient
framework to construct models of the Ω-Theory is the so-called Zermelo-Fraenkel-
Boffa set theory ZFBC which, roughly speaking, is a non-wellfounded variant of
ZFC where the axiom of foundation is replaced by other basic principles. Since
the discussion of the details of this theory is beyond the scope of this paper, we
remand to Chapter 7 of [1] for further references about the foundational framework
of Ω-Theory.
4.2 The Axioms of Ω-Theory
As we did in the case of Ω-Calculus, we begin the presentation of Ω-Theory by
stating all the axioms of the theory. In analogy to the first axiom of Ω-Calculus,
the first axiom of Ω-Theory is an existence axiom that ensures the existence of the
Ω-limit of every function that takes values in Ω.
66

4.2. THE AXIOMS OF Ω-THEORY
Ω-Axiom 1 (Existence Axiom). Every function ϕ : Pfin(Ω) → Ω has a unique
Ω-limit. This limit is denoted by
lim
λ↑Ω ϕ(λ)
Since we are no longer restricting the range of functions we consider, it’s useful to
introduce an axiom that helps us distinguish between limits of numbers and limits of
sets and that explicitely rules how the latter behave. In particular, we require that
the Ω-limits of functions whose values are atoms is an atom, and that the Ω-limit
of functions whose values are sets behave in analogy to Definition 1.7.1.
Ω-Axiom 2 (Atoms and Sets Axiom). If ϕ(λ) is an atom for all λ ∈ Pfin(Ω), then
limλ↑Ω ϕ(λ) is an atom. If ϕ(λ) is a nonempty set for all λ ∈ Pfin(Ω), then
lim ϕ(λ) = lim ψ(λ) : ψ(λ) ∈ ϕ(λ) ∀λ ∈ Pfin(Ω)
λ↑Ω λ↑Ω
(4.1)
Moreover, we ask that the Ω-limit of the function with constant value the empty set
is the empty set:
lim
λ↑Ω c∅(λ) = ∅
In analogy to Ω-Calculus, we want the Ω-limit to be compatible with the field
operations of R. To do so, we impose it with a new axiom.
Ω-Axiom 3 (Real Numbers Axiom). The set
R ∗
= lim ψ(λ) : ψ(λ) ∈ R ∀λ ∈ Pfin(Ω)
λ↑Ω
is a field and, if ϕ : Pfin(Ω) → R is eventually constant, then
Moreover, for all ϕ, ψ : Pfin(Ω) → R:
lim ϕ(λ) = r
λ↑Ω
limλ↑Ω ϕ(λ) + limλ↑Ω ψ(λ) = limλ↑Ω(ϕ(λ) + ψ(λ))
limλ↑Ω ϕ(λ) · limλ↑Ω ψ(λ) = limλ↑Ω(ϕ(λ) · ψ(λ))
As last axiom regarding Ω-limt, we need to introduce an axiom to rule that
Ω-limit is coherent with respect to composition of functions.
Ω-Axiom 4 (Composition Axiom). If ϕ, ψ : Pfin(Ω) → Ω satisfy
then also
lim ϕ(λ) = lim ψ(λ)
λ↑Ω λ↑Ω
lim f(ϕ(λ)) = lim f(ψ(λ))
λ↑Ω λ↑Ω
for all f for which are well defined the compositions f ◦ ϕ and f ◦ ψ.
67

CHAPTER 4. Ω-THEORY
4.3 Generalizing concepts
Once the axioms of the theory are given, we can generalize to Ω-Theory the definitions
and concepts introduced in Chapter 1 for the case of real subsets and functions.
A lot of new definitions will follow closely their counterparts, so we will give them
without many comments.
Definition 4.3.1. If E ⊆ Ω, we define
lim ϕ(λ) = lim ϕ(λ ∩ E)
λ↑E λ↑Ω
Definition 4.3.2. The natural extension of a set E ⊆ Ω is given by
E ∗ = lim
λ↑Ω cE(λ)
and, by Ω-Axiom 2, it’s equal to
E ∗
= lim ψ(λ) : ψ(λ) ∈ E ∀λ ∈ Pfin(Ω)
λ↑Ω
The hyperfinite extension of a set E ⊆ Ω is given by
E ◦ = lim
λ↑E λ
Definition 4.3.3. Let f : E → F . The natural extension of f, which we denote
f ∗ , is defined by:
f ∗ (lim
λ↑Ω ϕ(λ)) = lim
λ↑Ω f(ϕ(λ))
Notice that, by definition, f ∗ : E ∗ → F ∗ .
The hyperfinite extension of a function f : E → F is defined as the restriction
of f ∗ to E ◦ :
f ◦ = f ∗
|E ◦ : E◦ → F ∗
Notice that, thanks to Ω-Axiom 4, the natural extension of a function f is well
defined. It’s easy to see that, if E and F ⊆ R, definition 4.3.2 coincide with definitions
1.7.3 and 1.7.4, and definition 4.3.3 coincide with definition 1.8.2.
In analogy to Definition 1.4.3, we give the following
Definition 4.3.4. For all ϕ, ψ : Pfin(Ω) → Ω, we define Qϕ,ψ = {λ ∈ Pfin(Ω) :
ϕ(λ) = ψ(λ)}. We define also
Q =
Qϕ,ψ : lim
λ↑Ω ϕ(λ) = lim
λ↑Ω ψ(λ)
and, as we did before, we will say that a set Q ∈ Q is qualified, and we will call Q
the set of qualified sets.
Once again, it’s straightforward to check that, if ϕ and ψ assume real values,
then the two definitions coincide, so the common notation is justified. Now, we want
to show that Corollary 1.4.7 apply to arbitrary functions.
68

4.3. GENERALIZING CONCEPTS
Proposition 4.3.5. For all ϕ, ψ : Pfin(Ω) → Ω, limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ) if and
only if Qϕ,ψ ∈ Q.
Proof. One of the implications is true by definition, so we will focus on the other.
Notice also that, if ϕ and ψ assume real values almost everywhere, then the thesis
holds by Corollary 1.4.7.
Now, suppose that Qϕ,ψ ∈ Q. If we define the function f : Pfin(Ω) → {0, 1} in
the following way:
2 if λ ∈ Qϕ,ψ
f(λ) =
0 if λ ∈ Qϕ,ψ
then, by Corollary 1.4.7, we deduce that limλ↑Ω f(λ) = 0. Notice that, if the function
“{ }” defined by {}(x) = {x} is internal to Ω, then f can also be defined by the
formula
f(λ) = |{ϕ(λ)}△{ψ(λ)}|
and, by repeated use of Ω-Axiom 4,
lim f(λ) = lim (|{ϕ(λ)}△{ψ(λ)}|) = |{lim ϕ(λ)}△{lim ψ(λ)}|
λ↑Ω λ↑Ω λ↑Ω λ↑Ω
From this equality we conclude ϕ(λ) = ψ(λ).
If {} is not internal to Ω, call A the domain of {} and B = Ω\A. Moreover, pose
Ba = {x ∈ B : x is an atom} and Bs = {x ∈ B : x is a set}. With those definitions,
f is almost everywhere equal to the function g : Pfin(λ) → R defined a.e. as follows:
⎧
⎪⎨
|{ϕ(λ)}△{ψ(λ)}| if ϕ(λ) and ψ(λ) ∈ A
ϕ(λ) − ψ(λ) if ϕ(λ) and ψ(λ) ∈ Ba
g(λ) =
⎪⎩
|ϕ(λ)△ψ(λ)| if ϕ(λ) and ψ(λ) ∈ Bs
2 otherwise
Using Corollary 1.4.7 and with an argument similar to the previous part of the proof,
we can conclude.
This Proposition has several consequences, that we present here in the form of
corollaries.
Corollary 4.3.6. For all ϕ : Pfin(Ω) → Ω, limλ↑Ω ϕ(λ) is an atom or is a set.
Proof. Since we assume that the atoms are real numbers, define the function
0 if ϕ(λ) ∈ R
ψa(λ) =
1 if ϕ(λ) ∈ Rc Now, if limλ↑Ω ψa(λ) = 0 then the set A = {λ ∈ Pfin(Ω) : ϕ(λ) ∈ R} is qualified,
hence we can exhibit a function ψ : Pfin(Ω) → R such that Qϕ,ψ = A. This and
Proposition 4.3.5 is sufficient to conclude that limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ) ∈ R ∗ is an
atom. In the same spirit, if ψa(λ) = 1, then the set S = {λ ∈ Pfin(Ω) : ϕ(λ) ∈ R}
is qualified and, with the same reasoning, then limλ↑Ω ϕ(λ) is a set.
Thanks to this Corollary and to Proposition 4.3.5, we can easily deduce another
characterization of Q.
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CHAPTER 4. Ω-THEORY
Corollary 4.3.7. If we call
A =
and
B =
then Q = A = B.
Qϕ,ψ : lim
λ↑Ω ϕ(λ) = lim
λ↑Ω ψ(λ) and ϕ(λ), ψ(λ) ∈ R ∀λ ∈ Pfin(Ω)
Qϕ,ψ : lim
λ↑Ω ϕ(λ) = lim
λ↑Ω ψ(λ) and ϕ(λ), ψ(λ) ∈ R c ∀λ ∈ Pfin(Ω)
Proof. By Corollary 4.3.6, it should be clear that Q = A ∪ B. By Proposition 4.3.5,
it’s easy to see that A = B.
The last, and maybe more important consequence of Proposition 4.3.5 is that,
in analogy to what happened in the case of Ω-Calculus, Q is a nonprincipal fine
ultrafilter on Pfin(Ω).
Corollary 4.3.8. Q is a nonprincipal fine ultrafilter on Pfin(Ω).
Proof. The proof follows easily by the characterization of Q given by Corollary 4.3.7
and with the same arguments as in the proof of Proposition 1.4.8.
Notice that, by Ω-Theory, we could prove all the results about extensions of sets
and functions of Sections 1.7 and 1.8 for the case of arbitrary subsets of Ω and of
arbitrary functions f : A ⊆ Ω → Ω. Since the proofs are almost identical to those
already seen, we will omit them. The reader is warned that, when the need arises,
we will feel free to use the properties ruled in Sections 1.7 and 1.8 in this more
general interpretation.
4.4 Transfer Principle by Ω-Theory
The fundamental feature of the Ω-Theory is that all “elementary properties” are
preserved under Ω-limits. In fact, with respect to Ω-Calculus, a more general version
of the transfer principle holds, where arbitrary sequences of mathematical objects
(that is, numbers and sets) are considered.
Transfer Principle (general form). An “elementary property” P is satisfied by
elements ϕ1(λ), . . . ϕn(λ) for almost all λ if and only if P is satisfied by the Ω-limits
limλ↑Ω ϕ1(λ), . . . , ϕn(λ).
The proof of this result requires some logic knowledge, and for this reason, we
will give it in Chapter 5, after we’ve introduced the necessary logic formalism. In
this section, we want to show that, using the theory developed so far, we can prove
directly several important instances of Transfer. In particular, we will show that
equality and membership transfer under Ω-limit, as well as the set operations and
the notions of ordered tuples, relations and functions. As a consequence, if one
is not interested by the logical aspect of Ω-Theory, the results contained in this
70

4.4. TRANSFER PRINCIPLE BY Ω-THEORY
section should give sufficient instances of Transfer Principle to practice everyday
mathematics.
As a first fundamental result of Ω-Theory, we will prove that the above transfer
principle apply to both the “elementary” properties of equality and membership.
Theorem 4.4.1 (Transfer of equality and membership). For all ϕ, ψ : Pfin(Ω) → Ω,
1. ϕ(λ) = ψ(λ) a.e. ⇐⇒ limλ↑Ω ϕ(λ) = limλ↑Ω ψ(λ)
2. ϕ(λ) ∈ ψ(λ) a.e. ⇐⇒ limλ↑Ω ϕ(λ) ∈ limλ↑Ω ψ(λ)
The first part of the Theorem has already been proven in Proposition 4.3.5. To
prove this important theorem, we will need some preliminary results. At first, we
will prove one of the implications of transfer of membership: if ϕ(λ) ∈ ψ(λ) a.e.,
then limλ↑Ω ϕ(λ) ∈ limλ↑Ω ψ(λ).
Proposition 4.4.2. For all ϕ, ψ : Pfin(Ω) → Ω, if ϕ(λ) ∈ ψ(λ) a.e., then
limλ↑Ω ϕ(λ) ∈ limλ↑Ω ψ(λ).
Proof. At first, define
ϕ ′ (λ) =
ϕ(λ) if ϕ(λ) ∈ ψ(λ)
0 if ϕ(λ) ∈ ψ(λ)
Since ϕ = ϕ ′ a.e., if we prove that limλ↑Ω ϕ ′ (λ) ∈ limλ↑Ω ψ(λ) then, by transfer of
equality, we can conclude that the thesis holds. To prove this result, define the
function
ψ ′
ψ(λ)
(λ) =
{0}
if ϕ(λ) ∈ ψ(λ)
if ϕ(λ) ∈ ψ(λ)
By definition, ϕ ′ (λ) ∈ ψ ′ (λ) for all λ ∈ Pfin(Ω), and this implies limλ↑Ω ϕ ′ (λ) ∈
limλ↑Ω ψ ′ (λ). On the other hand, ψ ′ (λ) = ψ(λ) a.e. and, by transfer of equality,
limλ↑Ω ψ ′ (λ) = limλ↑Ω ψ(λ), so the thesis follows.
Thanks to Proposition 4.4.2, we will now show that Ω-Axiom 2 allows a weaker
formulation. This is the last result we need to settle before we prove the second part
of Theorem 4.4.1.
Lemma 4.4.3 (Sets Axiom - a.e. version). If ϕ(λ) is a set a.e., then
lim ϕ(λ) = lim ψ(λ) : ψ(λ) ∈ ϕ(λ) a.e.
λ↑Ω λ↑Ω
Proof. Choose a function η such that η is a set for all λ ∈ Pfin(Ω) and Qϕ,η ∈ Q. If
ψ(λ) ∈ ϕ(λ) a.e., then we have the relation
{λ ∈ Pfin(Ω) : ψ(λ) ∈ η(λ)} ⊇ Qϕ,η ∩ {λ ∈ Pfin(Ω) : ψ(λ) ∈ ϕ(λ)}
and, since the last two sets are qualified by hypothesis, by Corollary 4.3.8 we conclude
that also ψ(λ) ∈ η(λ) a.e. Thanks to Proposition 4.4.2, we conclude that
limλ↑Ω ψ(λ) ∈ limλ↑Ω η(λ) = limλ↑Ω ϕ(λ).
Conversely, let x ∈ limλ↑Ω ϕ(λ) = limλ↑Ω η(λ). Then, by Ω-Axiom 2, x =
limλ↑Ω ψ(λ) for a suitable function ψ such that ψ(λ) ∈ η(λ) for all λ ∈ Pfin(Ω).
Since Qϕ,η ∈ Q, it follows that ψ(λ) ∈ ϕ(λ) is true a.e., as we wanted.
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CHAPTER 4. Ω-THEORY
Now, we are ready to prove Theorem 4.4.1.
Proof of Theorem 4.4.1. We have already observed that the proof of the Transfer of
equality is the content of Proposition 4.3.5.
The first implication of Transfer of membership has been proven in Proposition
4.4.2. To show that the other implication holds, suppose that limλ↑Ω ϕ(λ) ∈
limλ↑Ω ψ(λ). This implies that limλ↑Ω ψ(λ) is a nonempty set, hence Proposition 4.3.5
ensures that ψ(λ) is a nonempty set a.e. By Lemma 4.4.3, it follows the existence
of a function η(λ) satisfying η(λ) ∈ ψ(λ) a.e. and limλ↑Ω η(λ) = limλ↑Ω ϕ(λ). By
Proposition 4.3.5, this implies Qϕ,η ∈ Q, hence ϕ(λ) ∈ ψ(λ) a.e., and this concludes
the proof.
Example 4.4.4. In Example 1.5.2 we have seen that the numerical property P (x)
saying “x ∈ E” does not transfer when E is infinite. Now, if we consider the
relation R(x, E) saying “x ∈ E”, then Theorem 4.4.1 ensures that R(ϕ(λ), Eλ) a.e.
if and only if R(limλ↑Ω ϕ(λ), limλ↑Ω Eλ). In particular, this implies that the relation
“x ∈ E” transfers to the relation “x ∗ ∈ E ∗ ”. The key consideration is that we need
to take the Ω-limit of all elements involved in the relation, not only numbers.
Remark 4.4.5. Thanks to Theorem 4.4.1, we can give an intuitive idea of why we
require the existence of non-wellfounded sets. We start by defining inductively the
von Neumann natural numbers
• 0 = ∅
• n + 1 = n ∪ {n} = {0, . . . , n}
From this definition, it’s easy to verify that limλ↑Ω |λ| = n(R) by showing that
lim{0,
. . . , |λ|} = {0, . . . , n(R)}
λ↑Ω
At this point, for all k ∈ N, we can set n(R) − k = limλ↑Ω |λ| − k, with the convention
that n − k = ∅ whenever k > n. Now, we can exhibit an infinite ∈-descending chain:
n(R) ∋ n(R) − 1 ∋ . . . ∋ n(R) − k ∋ n(R) − (k + 1) ∋ . . .
and the existence of this chain is in open contrast with foundation axiom of ZFC set
theory.
We now collect in the next theorem several instances of the Transfer principle of
basic set operations.
Theorem 4.4.6. For all ϕ, ψ : Pfin(Ω) → Ω
72
1. ϕ(λ) ⊆ ψ(λ) a.e. ⇐⇒ limλ↑Ω ϕ(λ) ⊆ ψ(λ)
2. η(λ) = ϕ(λ) ∪ ψ(λ) a.e. ⇐⇒ limλ↑Ω η(λ) = limλ↑Ω ϕ(λ) ∪ limλ↑Ω ψ(λ)
3. η(λ) = ϕ(λ) ∩ ψ(λ) a.e. ⇐⇒ limλ↑Ω η(λ) = limλ↑Ω ϕ(λ) ∩ limλ↑Ω ψ(λ)
4. η(λ) = ϕ(λ) \ ψ(λ) a.e. ⇐⇒ limλ↑Ω η(λ) = limλ↑Ω ϕ(λ) \ limλ↑Ω ψ(λ)

4.4. TRANSFER PRINCIPLE BY Ω-THEORY
5. η(λ) = (ϕ(λ), ψ(λ)) a.e. ⇐⇒ limλ↑Ω η(λ) = (limλ↑Ω ϕ(λ), limλ↑Ω ψ(λ))
6. η(λ) = ϕ(λ) × ψ(λ) a.e. ⇐⇒ limλ↑Ω η(λ) = limλ↑Ω ϕ(λ) × limλ↑Ω ψ(λ)
Proof. Part 2 follows from the following chain of implications: if η(λ) = ϕ(λ) ∪ ψ(λ)
a.e., then limλ↑Ω γ(λ) ∈ η(λ) if and only if γ(λ) ∈ η(λ) a.e. if and only if γ(λ) ∈ ϕ(λ)
a.e. or γ(λ) ∈ ψ(λ) a.e. if and only if limλ↑Ω γ(λ) ∈ limλ↑Ω ϕ(λ) or limλ↑Ω γ(λ) ∈
limλ↑Ω ψ(λ).
Similarly, if η(λ) = ϕ(λ) ∩ ψ(λ) a.e., then limλ↑Ω γ(λ) ∈ η(λ) if and only if
γ(λ) ∈ η(λ) a.e. if and only if γ(λ) ∈ ϕ(λ) a.e. and γ(λ) ∈ ψ(λ) a.e. if and only if
limλ↑Ω γ(λ) ∈ limλ↑Ω ϕ(λ) and limλ↑Ω γ(λ) ∈ limλ↑Ω ψ(λ), and this proves part 3.
The same argument can be used in the proof of part 4: if η(λ) = ϕ(λ)\ψ(λ) a.e.,
then limλ↑Ω γ(λ) ∈ η(λ) if and only if γ(λ) ∈ η(λ) a.e. if and only if γ(λ) ∈ ϕ(λ)
a.e. and γ(λ) ∈ ψ(λ) a.e. if and only if limλ↑Ω γ(λ) ∈ limλ↑Ω ϕ(λ) and limλ↑Ω γ(λ) ∈
limλ↑Ω ψ(λ).
To see that part 1 holds, notice that ϕ(λ) ⊆ ψ(λ) a.e. if and only if ϕ(λ)∪ψ(λ) =
ψ(λ) a.e. if and only if limλ↑Ω ψ(λ) = limλ↑Ω ϕ(λ) ∪ limλ↑Ω ψ(λ) if and only if
limλ↑Ω ϕ(λ) ⊆ limλ↑Ω ψ(λ).
Part 5 is a consequence of the following chain of equalities
lim(ϕ(λ),
ψ(λ))
λ↑Ω
= lim{{ϕ(λ)},
{ϕ(λ), ψ(λ)}}
λ↑Ω
= lim{ϕ(λ)},
lim{ϕ(λ),
ψ(λ)}
λ↑Ω λ↑Ω
= lim ϕ(λ)
λ↑Ω
, lim ϕ(λ), lim ψ(λ)
λ↑Ω λ↑Ω
=
Part 6 follows from part 5 and by Lemma 4.4.3.
lim ϕ(λ), lim ψ(λ)
λ↑Ω λ↑Ω
As a consequence of part 5 of Theorem 4.4.6, we deduce that Ω-limit is coherent
with n-tuples.
Corollary 4.4.7. For all n ∈ N,
lim(ϕ1(λ),
. . . , ϕn(λ)) =
λ↑Ω
lim ϕ1(λ), . . . , lim ϕn(λ)
λ↑Ω λ↑Ω
Proof. The proof is obtained by induction and by using part 5 of Theorem 4.4.6.
Thanks to the previous results, we can show that n-ary relations and functions
transfer under Ω-limit. Recall that a binary relation R is defined as a set of ordered
pairs, and the writing R(a, b) or aRb stands for (a, b) ∈ R. The domain and range
of R are defined by
dom(R) = {a : ∃b R(a, b)} and ran(R) = {b : ∃a R(a, b)}
Theorem 4.4.8 (Transfer of binary relations). Rλ is a binary relation satisfying
dom(Rλ) = Aλ and ran(Rλ) = Bλ for almost all λ ∈ Pfin(Ω) if and only if R =
limλ↑Ω Rλ is a binary relation with dom(R) = A = limλ↑Ω Aλ and ran(R) = B =
limλ↑Ω Bλ.
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CHAPTER 4. Ω-THEORY
Proof. Without loss of generality, we can suppose Rλ = ∅, Aλ = ∅ and Bλ = ∅ for
all λ ∈ Pfin(Ω).
To prove the first implication, given an arbitrary function η(λ) ∈ Rλ for all
λ ∈ Pfin(Ω), we have that η(λ) = (ϕ(λ), ψ(λ)) ∈ Aλ × Bλ for all λ ∈ Pfin(Ω)
such that Rλ is a relation, hence for almost all λ. By transfer of ordered pairs,
this implies limλ↑Ω η(λ) ∈ A × B. This proves that R is a binary relation with
dom(R) ⊆ A and ran(B) ⊆ B. To see that the inclusions are equalities, consider
a function ϕ(λ) ∈ Aλ and a function ϕ(λ) ∈ Bλ for all λ such that Rλ is a binary
relation. For those functions, we have
lim ϕ(λ), lim ψ(λ)
λ↑Ω λ↑Ω
= lim
λ↑Ω (ϕ(λ), ψ(λ)) ∈ R
In particular, limλ↑Ω ϕ(λ) ∈ dom(R), hence the inclusion A ⊆ dom(R). The inclusion
B ⊆ ran(R) can be proven with a similar argument.
To show that the second implication holds, define Q = {λ ∈ Pfin(Ω) : Rλ is
a binary relation}. For every λ ∈ Q, pick an element η(λ) ∈ Rλ which is not an
ordered pair. If Q was not qualified, then by Theorem 4.4.6 limλ↑Ω η(λ) would not
be an ordered pair, hence R would not be a binary relation. This proves that Q is
qualified, and so Rλ is a binary relation a.e.
Now, fix the family of sets {A ′ λ }λ∈Pfin(Ω) and {B ′ λ }λ∈Pfin(Ω) in a way such that
A ′ λ = dom(Rλ) and B ′ λ = ran(Rλ) whenever λ ∈ Q. By the first implication of
the theorem, we get dom(R) = limλ↑Ω A ′ λ and ran(R) = B′ λ
but, by hypothesis, we
already know dom(R) = limλ↑Ω Aλ and ran(R) = Bλ. By Transfer of equalities,
this implies Aλ = A ′ λ and Bλ = B ′ λ a.e., so we conclude that dom(Rλ) = Aλ and
ran(Rλ) = Bλ is true almost everywhere, as desired.
As the Transfer of ordered tuples follows from Transfer of ordered pairs, from
Theorem 4.4.8 we deduce that n-ary relations transfer under Ω-limit.
Corollary 4.4.9 (Transfer of n-ary relations). For all n ∈ N, Rλ is a n-ary relation
with dom(Rλ) = Aλ and ran(Rλ) = Bλ for almost all λ ∈ Pfin(Ω) if and only if
R = limλ↑Ω Rλ is a n-ary relation with dom(R) = limλ↑Ω Aλ and ran(R) = limλ↑Ω Bλ.
Proof. The proof is obtained by induction from the one of Theorem 4.4.8.
Another consequence of Theorem 4.4.8 is that functions transfer under Ω-limits.
Theorem 4.4.10 (Transfer of functions). fλ is a function fλ : Aλ → Bλ for almost
all λ ∈ Pfin(Ω) if and only if f = limλ↑Ω fλ : limλ↑Ω Aλ → limλ↑Ω Bλ is a function.
Proof. For a matter of commodity, define A = limλ↑Ω Aλ and B = limλ↑Ω Bλ.
Since a function can be seen as a particular relation, by Theorem 4.4.8 we
conclude that, if fλ : Aλ → Bλ is a function for almost all λ ∈ Pfin(Ω), then
f = limλ↑Ω fλ is a relation with dom(f) = limλ↑Ω Aλ and ran(f) = limλ↑Ω Bλ. It
remains only to check that, for all a ∈ A, there exists only one b ∈ B such that
(a, b) ∈ f. To do so, for a given a ∈ A consider a function ϕ(λ) ∈ Aλ for all λ
such that fλ is a function, and satisfying limλ↑Ω ϕ(λ) = a. Now, suppose there are
74

4.5. A MODEL FOR Ω-THEORY
b = limλ↑Ω ψ(λ) and b ′ = limλ↑Ω ψ ′ (λ) such that (a, b) ∈ f and (a, b ′ ) ∈ f. By Theorems
4.4.1 and 4.4.6, this means that both (ϕ(λ), ψ(λ)) ∈ fλ and (ϕ(λ), ψ ′ (λ)) ∈ fλ
are true almost everywhere. By the fact that fλ are functions almost everywhere, we
conclude ψ(λ) = ψ ′ (λ) almost everywere and, by Theorem 4.4.1, this ensures b = b ′ ,
as we wanted.
To prove the other implication, thanks to Theorem 4.4.8 it’s sufficient to prove
that for almost all λ, if (aλ, bλ) ∈ fλ and (aλ, b ′ λ ) ∈ fλ, then bλ = b ′ λ . Suppose that
this is not the case, and that for almost every λ there are aλ ∈ Aλ and bλ = b ′ λ ∈ Bλ
satisfying (aλ, bλ) ∈ fλ and (aλ, b ′ λ ) ∈ fλ. By Theorem 4.4.8, we would then have
(limλ↑Ω aλ, limλ↑Ω bλ) ∈ f and (limλ↑Ω aλ, limλ↑Ω b ′ λ ) ∈ f. Since bλ = b ′ λ a.e., also
, but this contradicts the fact that f is a function.
limλ↑Ω bλ = limλ↑Ω b ′ λ
Notice that Theorem 4.4.10 ensures that Definition 4.3.3 of f ∗ is coherent with
the definition of f ∗ as the Ω-limit of its graph. With similar arguments as in the
proof of Theorem 4.4.10, it can also be shown that f is injective (resp. surjective)
if and only if fλ is injective (resp. surjective) for almost all λ.
We want to explicitely observe that the only basic set operation that is not
preserved under Ω-limits is the powerset. In Example 5.1.6, after we’ve formally
defined what an “elementary property” is, we will give an intuitive explanation of
why it is so.
4.5 A model for Ω-Theory
Once again, in order to prove the consistency of Ω-Theory, we find opportune to
exhibit a model for Ω-Theory. We recall that in Section 1.11 we obtained a model
for Ω-Theory from a quotient of the ring Fun(Pfin(Ω), R) by an appropriate maximal
ideal. To give a model for Ω-Theory, we want to extend this process to the
set Fun(Pfin(Ω), Ω). This set doesn’t have always a natural structure of ring, like
Fun(Pfin(Ω), R) had, so we will need to use a different approach that takes into
account those differences. The idea is to use ultrapowers, an algebraic construction
based on ultrafilters. Ultrapowers are, so to speak, the generalization to arbitrary
sets of the concept of quotient of a ring by a maximal ideal. Before we can give a
precise meaning to this intuitive statement, we find opportune to define a relation
that extends the notion of two functions ϕ, ψ : Pfin(Ω) → R being equal on a
qualified set. More formally, we give the following definition:
Definition 4.5.1. Let E, F be two sets and U an ultrafilter on E. We define a
relation ∼U on Fun(E, F ) in the following way: ϕ ∼U ψ if and only if Qϕ,ψ ∈ U.
Notice that this definition is independent on the algebraic structure of Fun(E, F ).
As usual, we leave to the reading the easy checkings that ∼U is an equivalence
relation.
In the same way as Definition 4.5.1 extends the notion of two functions being
equal on a qualified set, the notion of ultrapower extends the notion of the quotient
of a ring by a maximal ideal.
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CHAPTER 4. Ω-THEORY
Definition 4.5.2. Let E, F be sets and U an ultrafilter on E. The ultrapower of F
modulo U on E is the quotient set
F E U = Fun(E, F )/ ∼U
Sometimes, when the set E is fixed or when there will be no ambiguity, we will use
the notation FU instead of F E U .
Remark 4.5.3. Given an ultrapower F E U
defined in the following way:
J : Fun(E, F ) → F E U
, there exists a “natural” map
J(ϕ) = [ϕ] = {ψ ∈ Fun(E, F ) : ϕ ∼U ψ}
In other words, J is the canonical projection from Fun(E, F ) to F E U
surjective.
. Clearly, J is
If Fun(E, F ) is a ring, then the notion of ultrapower of F modulo U is equivalent
to the one of quotient of Fun(E, F ) modulo the maximal ideal mU introduced in
Proposition 1.10.14.
Proposition 4.5.4. If Fun(E, F ) is a commutative ring and U is an ultrafilter on
E, then
F E U ≡ Fun(E, F )/mU
Proof. By definition, F E U
= {[ϕ] : ϕ ∈ Fun(E, F )}, where
[ϕ] = {ψ ∈ Fun(E, F ) : ϕ ∼U ψ}
On the other hand, Fun(E, F )/mU = {[[ϕ]] : ϕ ∈ Fun(E, F )}, where
[[ϕ]] = {ψ ∈ Fun(E, F ) : ϕ − ψ ∈ mU}
Explicitating the definition of mU, we see that
[[ϕ]] = {ψ ∈ Fun(E, F ) : Qϕ−ψ,c0 ∈ U} = {ψ ∈ Fun(E, F ) : ϕ ∼U ψ} = [ϕ]
and this proves that F E U = Fun(E, F )/mU as sets. Now, we only need to prove that,
if ϕ, ψ ∈ Fun(E, F ), then the operations in F E U and in Fun(E, F )/mU are compatible
or, in other words, that in F E U , [ϕ + ψ] = [ϕ] + [ψ]. By definition,
[ϕ] + [ψ] = {ξ + ζ : Qξ,ϕ and Qζ,ψ ∈ U}
Since U is an ultrafilter, we have that Qξ+ζ,ϕ+ψ = Qξ,ϕ∩Qζ,ψ ∈ U, so ξ+ζ ∈ [ϕ+ψ] or,
in other words, [ϕ]+[ψ] ⊆ [ϕ+ψ]. On the other hand, it’s clear that ϕ+ψ ∈ [ϕ]+[ψ],
and this implies [ϕ + ψ] ⊆ [ϕ] + [ψ]. We can conclude [ϕ + ψ] = [ϕ] + [ψ], as we
wanted. A similar argument can be used to prove that [ϕ · ψ] = [ϕ] · [ψ], and this
concludes the proof.
76

4.5. A MODEL FOR Ω-THEORY
In our concrete case, we want to obtain a model for Ω-Theory by considering an
appropriate ultrapower of Ω modulo an ultrafilter on Pfin(Ω). To do so, we only
need to define a suitable ultrafilter UΩ over Pfin(Ω) in a way that all the axioms of
Ω-Theory are satisfied by the map J in the corresponding ultrapower ΩUΩ . We find
opportune to summarize all the properties of Ω-Theory we need to satisfy, and the
relative restrictions imposed on UΩ:
• To satisfy Ω-Axiom 1, we don’t have to impose any property on UΩ: since the
projection from Fun(Pfin(Ω), Ω) to ΩUΩ is surjective, the axiom will hold.
• The same applies for Ω-Axiom 2: if ϕ(λ) is an atom for all λ ∈ Pfin(Ω), this
means that ϕ : Pfin(Ω) → R, so its Ω-limit will belong to R ∗ , as desired. The
same reasoning applies if ϕ(λ) is a set for all λ ∈ Pfin(Ω). It only remains
to show that equation 4.1 holds, and we will impose its validity during the
construction of the model. For a thorough presentation of the proof method
we will use to show that Ω-Axiom 2 holds in the model, we remand to Chapter
6 of [1].
• As for Ω-Axiom 3, by Theorem 1.9.9 it’s sufficient that UΩ is a nonprincipal
fine ultrafilter.
• In order to satisfy Ω-Axiom 4, it’s sufficient that, for all f ∈ Fun(E, F ) and
for all ϕ, ψ : Pfin(Ω) → E, if limλ↑Ω ϕ = limλ↑Ω ψ then [f ◦ ϕ] = [f ◦ ψ] as
elements of ΩUΩ . This request is satisfied once we prove that Qf◦ϕ,f◦ψ ∈ Q,
and this will follow from the hypothesis Qϕ,ψ ∈ Q.
Putting everything together, we require only that UΩ is a nonprincipal fine ultrafilter.
Theorem 4.5.5. Ω-Axioms 1, 2, 3 and 4 hold in the ultrapower ΩUΩ .
Proof. Since the map J : Fun(Pfin(Ω), Ω) → ΩUΩ is surjective, Ω-Axiom 1 holds.
Ω-Axiom 3 holds as a consequence of Theorem 1.9.9, Proposition 1.10.14 and
Proposition 4.5.4.
As for Ω-Axiom 4, pick E, F ⊆ Ω and a function f : E → F . We need to prove
that, if ϕ and ψ ∈ Fun(Pfin(Ω), E) satisfy [ϕ] = [ψ] as elements of ΩUΩ , then it’s also
true that [f ◦ ϕ] = [f ◦ ψ] as elements of ΩUΩ . The hypothesis [ϕ] = [ψ] is equivalent
to Qϕ,ψ ∈ UΩ, but this implies that also Qf◦ϕ,f◦ψ ∈ UΩ, so when we consider their
class of equivalence in the ultrafilter, we obtain [f ◦ ϕ] = [f ◦ ψ].
Now, we will impose that Ω-Axiom 2 holds in the model. We start by defining
A = {x ∈ Ω : x is an atom} and by considering the projection
J0 : Fun(Pfin(Ω), A) → AUΩ
defined so that J0([cr]) = r for all r ∈ R. Notice that the request that the Ω-limit of
an atom must be an atom translates to limλ↑Ω cA(λ) = A. This, in particular, means
that the Ω-limit of constant functions must be a bijection between A and AUΩ , and
that every r ∈ R is a fixed point of the Ω-limit. To fulfill this condition, we impose
that A satisfy the additional hypothesis |A| |Ω| = |A| (notice that this can be done
77

CHAPTER 4. Ω-THEORY
without loss of generality by adding some “dummy” atoms that won’t be used in
the working practice). From this hypothesis we deduce the following equation:
|A| ≤ |AUΩ | ≤ |Fun(Pfin(Ω), A)| = |A| |Ω| = |A|
and, in particular, that |A| = |AUΩ |. As a consequence, we can pick a bijection
ψ0 : AUΩ → A such that ψ([ϕ]) = J0(ϕ) for all ϕ ∈ Fun(Pfin(Ω), R). This is
sufficient to satisfy the first part of Ω-Axiom 2.
At this point, for all ϕ ∈ Fun(Pfin(Ω), A), set limλ↑Ω ϕ(λ) = ψ0(ϕ). Now, in the
model, we define
lim ∅ = ∅
λ↑Ω
and, for all Bλ ⊆ A,
lim Bλ =
λ↑Ω
lim ϕ(λ) : ϕ(λ) ∈ Bλ for all λ ∈ Pfin(Ω)
λ↑Ω
and we continue defining the Ω-limit of sets by (possibly transfinite) induction. This
definition ensures that the second part of Ω-Axiom 2 holds.
78

Chapter 5
The formal version of Transfer
Principle
It seems that if we want to be able
to communicate at all, we have to
adopt some common base, and it
pretty well has to include logic.
Douglas R. Hofstadter
Probably, Transfer Principle is the most important theorem that Ω-Theory
can prove. For this reason, we find interesting to give it a precise formalization
inside first order logic for set theory. To do so, we need to introduce
the logic formalism for set theory and to give a rigorous definition the concept of
“elementary property”. Then, after some interesting examples, we will finally show
that Ω-Theory proves that every elementary property is preserved under Ω-limit.
5.1 Logic formalism for set theory
It’s a well known fact that the theory of real numbers can be developed inside set
theory: all we need to describe their properties and behaviours is the language
of set theory and, in particular, the language of first-order logic plus the equality
and membership relations. We begin the introduction of the logic formalism for set
theory by specifying the “alphabet” of symbols of the first-order language of set
theory, which will be used to construct our formulas.
• First of all, we require that our alphabet has a denumerable supply of variables,
that we will denote by x, y, z, . . . , x1, x2, . . .
• Then, we ask that our language has all the logic connectives of predicate logic:
– Negation: ¬ (not).
– Conjunction: ∧ (and).
– Disjunction: ∨ (or).
79

CHAPTER 5. THE FORMAL VERSION OF TRANSFER PRINCIPLE
– Implication: ⇒ (if . . . then).
– Double implication: ⇔ (if and only if).
• In the same spirit, we ask that our language has all the quantifiers of first
order logic:
– Existential quantifier: ∃ (there exists).
– Universal quantifier: ∀ (for all).
• We require also that our language has the equality symbol =.
• Since this is the alphabet of the language of set theory, we need also the symbol
∈ of membership.
• At last, we include the parentheses “(” and “)”.
Using this language, we can now give the formal definition of an elementary
formulas. As with many other definitions in logic, the elementary formulas will be
defined as the transitive closure of the elementary atomic formulas under certain
operations.
Definition 5.1.1. An elementary formula is a finite string of symbols in the above
language in which variables are distinguished into free variables and bound variables,
and which are obtained according to the following rules:
EF 1 (Atomic Formulas). Let x and y be variables. Then
(x = y), (x ∈ y)
are elementary formulas, named atomic formulas, where:
• the free variables are x and y;
• there are no bound variables.
EF 2 (Restricted quantifiers). Let σ be an elementary formula and assume that
1. the variable x is free in σ,
2. the variable y is not bound in σ.
Then also
are elementary formulas, where
80
(∀x ∈ y) σ, (∃x ∈ y) σ
• the free variables are y along with the free variables of σ;
• the bound variables are x along with the bound variables of σ.

5.1. LOGIC FORMALISM FOR SET THEORY
EF 3 (Negation). Let σ be an elementary formula. Then also
is an elementary formula, where
(¬σ)
• the free variables are the same as in σ;
• the bound variables are the same as in σ.
EF 4 (Binary logic connectives). Let σ and τ be elementary formulas, and assume
that:
1. every free variable in σ is not bound in τ;
2. every free variable in τ is not bound in σ.
Then also
are elementary formulas, where
(σ ∧ τ), (σ ∨ τ), (σ ⇒ τ), (σ ⇔ τ)
• the free variables are union of the free variables of σ and the free variables of
τ;
• the free variables are union of the bound variables of σ and the bound variables
of τ.
A few comments are in order. Clearly, in the inductive process of forming an
elementary formula, the first rule to be applied is EF1: in other words, every formula
is built on atomic formulas (and this justifies the name “atomic”). An arbitrary
formula is then obtained from atomic formulas by finitely many iterations of the
other three rules, in whatever order.
The restricted quantifiers rule EF2 is the only one that produces bound variables.
The idea is that a variable is bound when it is quantified. We remark that a variable
can be quantified only if it is free in the given formula, i.e. only if it actually appears
and it has been not quantified already.
It is worth remarking that quantifications are only permitted in the restricted
forms (∀x ∈ y) or (∃x ∈ y), where the “scope” of the quantified variable x is
“restricted” by another variable y. In order to avoid potential ambiguities, it is also
required that the “bounding” variable y do not appear bound itself in the given
formula.
The negation rule EF3 is the simplest one, as it applies to all formulas without
any restrictions, and it produces new formulas having the same free and bound
variables as the starting ones.
Also the binary connectives rule EF4 is quite simple, and it produces formulas
where the free and the bound variables are simply obtained by putting together the
free and the bound variables of the two given formulas, respectively. The provisions
in rule EF4 are given to prevent “connecting” two formulas where a same variable
appears free in one of them, and bound in the other.
For a matter of commodity, we will follow the common use and adopt familiar
short-hands to simplify notation. For instance:
81

CHAPTER 5. THE FORMAL VERSION OF TRANSFER PRINCIPLE
• we will write “x = y” instead of “¬(x = y)”;
• we will write “x ∈ y” instead of “¬(x ∈ y)”;
• we will write “∀x1, . . . , xn ∈ y σ” instead of “(∀x1 ∈ y) . . . (∀xn ∈ y) σ”;
• we will write “∃x1, . . . , xn ∈ y σ” instead of “(∃x1 ∈ y) . . . (∃xn ∈ y) σ”.
Moreover, we will take the freedom of using parentheses informally, and omit some
of them whenever confusion is unlikely. So, we may write “∀x ∈ y σ” instead of the
correct “(∀x ∈ y) σ”; or “σ ∧ τ” instead of “(σ ∧ τ)”, and so on.
Another agreement is that the negations ¬ bind more strongly than conjunctions
∧ and disjunctions ∨, which in turn bind more strongly than implications ⇒ and
double implications ⇔. So, for example,
• we may write “¬σ ∧ τ” instead of “((¬σ) ∧ τ)”;
• we may write “¬σ ∧ τ ⇒ ρ” instead of “(((¬σ) ∧ τ) ⇒ ρ)”;
• we may write “σ ⇒ τ ∧ ρ” instead of “(σ ⇒ (τ ∧ ρ))”.
An important notation is the following: when writing
σ(x1, . . . , xn)
we will mean that x1, . . . , xn are all and only the free variables that appear in the
formula σ. The intuition is that the truth or falsity of a formula depends only on the
values given to its free variables, whereas bound variables can be renamed without
changing the meaning of the formula.
We are now ready for the fundamental definition:
Definition 5.1.2 (Property expressed in elementary form). A property of mathematical
objects A1, . . . , An is expressed in elementary form if it is written down by
taking an elementary formula σ(x1, . . . , xn) and and by replacing all occurrences of
each free variable xi with Ai. In this case, we will denote
σ(A1, . . . , An)
and the objects A1, . . . , An are referred to as constants.
By a slight abuse, we will often say elementary property instead of the correct
“property expressed in elementary form”.
The motivation of our definition is the well-known fact that virtually all properties
considered in mathematics can be formulated in elementary form. Below is a
list of examples that include the fundamental ones.
Example 5.1.3. Each property is followed by one of its possible expressions in
elementary form. For simplicity, in each item we use short-hands for properties that
have been already expressed in elementary form in a previous case.
82
• “A ⊆ B”: (∀x ∈ A)(x ∈ B);

5.1. LOGIC FORMALISM FOR SET THEORY
• “C = A ∪ B”: (A ⊆ C) ∧ (B ⊆ C) ∧ (∀x ∈ C)(x ∈ A ∨ x ∈ B);
• “C = A ∩ B”: (C ⊆ A) ∧ (∀x ∈ A)(x ∈ B ⇔ x ∈ C);
• “C = A \ B”: (C ⊆ A) ∧ (∀x ∈ A)(x ∈ B ⇔ x ∈ C);
• “C = {a1, . . . , an}”: (a1 ∈ C)∧. . .∧(an ∈ C)∧(∀x ∈ C)(x = a1∨. . .∨x = an);
• “{a1, . . . , an} ∈ C”: (∃x ∈ C)(x = {a1, . . . , an});
• “C = (a, b)”: ({a} ∈ C) ∧ ({a, b} ∈ C) ∧ (∀x ∈ C)(x = {a} ∨ x = {a, b});
• In the same way, the formula for “C = (a1, . . . , an)” can be obtained from the
previous one by using the recursive definition of ordered n-tuple;
• “(a1, . . . , an) ∈ C”: (∃x ∈ C)(x = (a1, . . . , an));
• “C = A1 × . . . × An”: (∀x1 ∈ A) . . . (∀xn ∈ An)((x1, . . . , xn) ∈ C) ∧ (∀z ∈
C)(∃x1 ∈ A1) . . . (∃xn ∈ An)(z = (x1, . . . , xn));
• “R is a n-place relation on A”: (∀z ∈ R)(∃x1, . . . , xn ∈ A)(z = (x1, . . . , xn));
• “f : A → B”: (f ⊆ A × B) ∧ (∀a ∈ A)(∃b ∈ B)((a, b) ∈ f) ∧ (∀a ∈ A)(∀b, b ′ ∈
B)((a, b), (a, b ′ ) ∈ f ⇒ b = b ′ );
• “f(a1, . . . , an) = b”: ((a1, . . . , an), b) = (a1, . . . , an, b) ∈ f;
• “x < y in R”: (x, y) ∈ R where R ⊆ R × R is the order relation on R.
Remark 5.1.4. When expressing a property in elementary form, quantifiers can
only be used in the restricted forms:
“(∀x ∈ A) σ(x, . . .)” and “(∃x ∈ X) σ(x, . . .)”
So, whenever quantifying on a variable x, it must be always specified the set A where
the variable x is ranging. Two crucial examples are the following:
• Quantification ranging on subsets:
must be reformulated
respectively;
• Quantification ranging on functions:
must be reformulated
respectively.
“∀x ⊆ X . . . ” or “∃x ⊆ X . . . ”
“∀x ∈ P(X) . . . ” and “∃x ∈ P(X) . . . ”
“∀f : A → B . . . ” or “∃f : A → B . . . ”
“∀f ∈ Fun(A, B) . . . ” and “∃f ∈ Fun(A, B) . . . ”
83

CHAPTER 5. THE FORMAL VERSION OF TRANSFER PRINCIPLE
Example 5.1.5. An elementary formulation of the completeness property of the
reals requires the powerset P(R) as a constant:
∀X ∈ P(R)(“X bounded above” ⇒ (∃x ∈ R)“x = sup A”)
Where “X bounded above” and “x = sup A” are the short-hands for the corresponding
expressions in elementary form:
• (∃x ∈ R)(∀y ∈ X)(y < x), and
• (∀y ∈ X)(y ≤ x) ∧ (∀z ∈ R)((∀t ∈ X)(t ≤ z)) ⇒ x ≤ z,
respectively.
It is worth remarking that a same property may be expressed both in an elementary
form and in a non-elementary form. The typical example involves the powerset
operation.
Example 5.1.6. “P(A) = B” is trivially an elementary property of constants P(A)
and B, but cannot be formulated as an elementary property of constants A and
B. In fact, while the inclusion B ⊆ P(A) is formalized in elementary form by
“(∀x ∈ B)(∀y ∈ x)(y ∈ A)”, the other inclusion P(A) ⊆ B does not admit any
elementary formulation with A and B as constants. The point is that, under our
rules, quantifications over subsets “(∀x ⊆ A)(x ∈ B)” are not allowed.
5.2 Transfer Principle
Finally, we have everything we need to show that the Transfer Principle holds in the
Ω-Theory or, in other words, that all properties expressed in elementary form are
preserved under Ω-limits. This theorem is the fundamental result of Ω-Theory.
Theorem 5.2.1 (Transfer Principle). Let σ(x1, . . . , xn) be an elementary formula
and ϕ1, . . . , ϕn : Pfin(Ω) → Ω. Then
σ(ϕ1(λ), . . . ϕn(λ)) a.e. ⇐⇒ σ lim ϕ1(λ), . . . , lim ϕn(λ)
λ↑Ω λ↑Ω
Proof. We proceed by induction on the complexity of the formulas. For atomic
formulas (x1 = x2) and (x1 ∈ x2), the thesis is given by Theorem 4.4.1.
The negation case ¬σ follows from the inductive hypothesis on σ and the property
that a set is qualified if and only if its complement is not qualified. Formally, we
have the following chain of equivalences:
¬σ(ϕ1(λ), . . . , ϕn(λ)) a.e. ⇐⇒ σ(ϕ1(λ), . . . , ϕn(λ)) not a.e.
⇐⇒ not σ (limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕn(λ))
⇐⇒ ¬σ (limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕn(λ))
The conjunction case (σ ∧ σ ′ ) is proved similarly as above, by using the property
that two sets are both qualified if and only if their intersection is qualified. The
cases of the other binary connectives (σ ∨ σ ′ ) and (σ ⇒ σ ′ ) and (σ ⇔ σ ′ ) can then
be proved by using the tautologies
84

• (σ ∨ σ ′ ) ≡ ¬(¬σ ∧ ¬σ ′ );
• (σ ⇒ σ ′ ) ≡ (¬σ ∨ σ ′ );
• (σ ⇔ σ ′ ) ≡ (σ ⇒ σ ′ ) ∧ (σ ′ ⇒ σ).
5.2. TRANSFER PRINCIPLE
The only cases left are the existential and unversal quantifiers. We will prove
the inductive step for the former, since the latter can be treated by considering the
logic equivalence
(∀x ∈ y)σ ≡ ¬(∃x ∈ y)¬σ
First, assume that the set Q = {λ : ∃x ∈ ψ(λ) τ(x, ϕ1(λ), . . . , ϕn(λ))} is qualified.
We can then find a function η : Pfin(Ω) → Ω satisfying the conditions that for
all λ ∈ Q, both η(λ) ∈ ψ(λ) and τ(η(λ), ϕ1(λ), . . . , ϕn(λ)) are true. Then, by the
inductive hypothesis, we have
lim η(λ) ∈ lim ψ(λ) and τ
λ↑Ω λ↑Ω
from which we deduce immediately
∃x ∈ lim
λ↑Ω ψ(λ) τ
lim η(λ), lim ϕ1(λ), . . . , lim ϕn(λ)
λ↑Ω λ↑Ω λ↑Ω
x, lim
λ↑Ω ϕ1(λ), . . . , lim
λ↑Ω ϕn(λ)
Conversely, if ∃x ∈ limλ↑Ω ψ(λ) τ (x, limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕn(λ)) then, by Transfer
of membership, there exists η : Pfin(Ω) → Ω such that η(λ) ∈ ψ(λ) a.e. and
τ (limλ↑Ω η(λ), limλ↑Ω ϕ1(λ), . . . , limλ↑Ω ϕn(λ)). By the inductive hypothesis, we deduce
τ(η(λ), ϕ1(λ), . . . , ϕn(λ)) a.e., and this concludes the proof.
As a direct consequence, we obtain that every property expressed in elementary
form holds for given objects A1, . . . An if and only if it holds for the corresponding
Ω-limits limλ↑Ω cA1(λ), . . . , limλ↑Ω cAn(λ). For a matter of commodity, if we define
limλ↑Ω cA(λ) = A ∗ , then this result can be stated as follows:
Corollary 5.2.2 (Transfer principle for hyper-images). Let σ(x1, . . . , xn) an elementary
formula. Then, for all A1, . . . , An,
σ(A1, . . . , An) ⇐⇒ σ(A ∗ 1, . . . , A ∗ n)
Proof. This result follows from Theorem 5.2.1 and by noticing that, for all λ ∈
Pfin(Ω),
σ(A1, . . . , An) ⇐⇒ σ (cA1(λ), . . . , cAn(λ))
In Example 5.1.3, we have seen that C = A ∪ B, C = A ∩ B, C = A \ B,
C = (a, b), C = A × B can all be expressed in elementary form. So, by Transfer
Principle, we obtain again the thesis of Theorem 4.4.6.
Similarly, the reader could re-prove all preservation properties under Ω-limits
presented in Chapter 4, by simply putting them in elementary form and then applying
Transfer. It is worth remarking that restricting to elementary sentences is not
85

CHAPTER 5. THE FORMAL VERSION OF TRANSFER PRINCIPLE
a limitation, because all mathematical properties can be rephrased in elementary
terms.
Plenty of applications of Transfer Principle can be easily conceived by the reader.
Some of them are given below as examples.
Example 5.2.3. If (A, 1”. Notice that the above sentence does not express the Archimedean
property of R ∗ , because the hypernatural number ν could be infinite.
By using correctly Transfer Principle, the interested reader could easily prove
the following Lemma, that will be used in Section 7.5:
Lemma 5.2.6. If ξ ∈ R ∗ , ξ ∼ 0 and ξ = 0, then there exists ν ∈ N ∗ satisfying
1/ν < ξ.
An interesting example of the use of this Lemma is given by the density of Q ∗
in R ∗ :
Example 5.2.7. Thanks to Lemma 5.2.6, one can deduce the density of Q ∗ in R ∗
with respect to the metric given by d ∗ (x, y) = |x − y|. Notice that this metric is the
natural extension of the metric d : R × R → R.
86

Chapter 6
An application: Non-Archimedean
Probability
The theory of probability as a
mathematical discipline can and
should be developed from axioms
in exactly the same way as
geometry and algebra.
A. Kolmogorov
One among the many interesting application of the Ω-Theory seems to be
that of Non-Archimedean Probability (that sometimes we will call NAP).
There are many approaches to the topic, and we signal [12], that use
the tools of Ω-limit to construct suitable NAP spaces that model some interesting
theorical situations (e.g. a fair lottery over N, Q and R among the others).
We already have a theory of Ω-limit, so we will follow a slightly different route:
in the first section, we will recall the axiomatic formulation of Kolmogorov’s probability
theory and then, after a brief discussion of its weaknesses, we will give an
axiomatization of Non-Archimedean Probability that uses the Ω-limit as a replacement
for the Conditional Probability Principle. Once we have defined the axioms
of NAP, we will begin to study their consequences and show some interesting examples.
In particular, we will show that we can interpret the idea of NAP spaces inside
Ω-Theory.
6.1 Kolmogorov’s axioms of probability
We start by recalling the definition of σ-algebra and then by giving an axiomatic
formulation of a probability over a generic set Υ that is equivalent to Kolmogorov’s
original axioms. The set Υ is usually called the sample space, and its elements
represent “elementary events”. Contrary to common use, we will not denote the
sample space by the letter Ω, since in our theory Ω is already used with a different
meaning.
Definition 6.1.1. A(Υ) is a σ-algebra over Υ if and only if
87

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
• A(Υ) ⊆ P(Υ);
• Υ ∈ A(Υ);
• A ∈ A(Υ) implies A c ∈ A(Υ);
• For every family {Ai}i∈N, Ai ∈ A(Υ) for all i ∈ N implies
i∈I Ai ∈ A(Υ);
• If A1, A2 ∈ A(Υ), then A1 ∩ A2 ∈ A(Υ).
An element A ∈ A(Υ) is called an event. If A is a singleton, then it’s called an
elementary event.
KA 1 (Domain and Range). The events are the elements of A(Υ), a σ-algebra over
Υ, and the probability is a function
P : A(Υ) → R
KA 2 (Positivity). For all A ∈ A(Υ), P (A) ≥ 0.
KA 3 (Normalization). P (Υ) = 1.
KA 4 (Additivity). For all A, B ∈ A(Υ), if A ∩ B = ∅, then P (A ∪ B) = P (A) +
P (B).
Definition 6.1.2. For all A, B ∈ A(Υ), B = ∅, we define the conditional probability
P (A|B) by
P (A ∩ B)
P (A|B) =
P (B)
KA 5 (Conditional Probability Principle). Let {An}n∈N be a family of events satisfying
the two conditions An ⊆ An+1 for all n ∈ N and Υ =
n∈N An. Then,
eventually P (An) > 0 and, for any B ∈ A(Υ),
P (B) = lim
n→∞ P (B|An)
An immediate consequence of those axioms are, for instance, the fact that 0 ≤
P (A) ≤ 1 for all A ∈ A(Υ), the monotony of probability and countable additivity.
Example 6.1.3. If the sample space is finite, then it’s usual to assume A(Υ) =
P(Υ). In this case, to define a probability over P(Υ) it’s sufficient to define a
normalized probability function on the elementary events, namely a function p :
Υ → R satisfying
pυ = 1
υ∈Υ
Once p is given, we can define a probability function P : P(Υ) → [0, 1] by posing
P (A) =
p(υ) (6.1)
υ∈A
It’s trivial to check that P satisfies all of Kolmogorov’s axioms.
88

6.2. NON-ARCHIMEDEAN PROBABILITY
Remark 6.1.4. Equation 6.1 cannot be generalized to the infinite case. In fact, if
Υ = R, then an infinite sum is well defined only if the set {x ∈ R : p(x) = 0} is
denumerable or finite.
Remark 6.1.5. The choiche of the range of probability together with countable
additivity may lead to some problems when the sample space is infinite.
A peculiarity of Kolmogorov’s probability is that, in general, A(Υ) = P(Υ). The
most known example is the probability measure given by Lebesgue measure, which
cannot be defined for all the sets in P(Υ). This seems an innocent feature, but it’s
equivalent to say that there are sets in P(R) which are not events (namely elements
of A(R)), even when they are the union of elementary events.
Another weak point for Kolmogorov’s probability is the interpretation of P (A) =
0 and P (A) = 1. Explicitely, there are plenty of situations in which we can find
some sets {Ai}i∈I such that, for all i ∈ I,
and
P
P (Ai) = 0 (6.2)
i∈I
Ai
= 1 (6.3)
This situation is very common when I is not denumerable. In this case, it looks as
if equation 6.2 states that each event Ai is impossible, whereas equation 6.3 states
that one of them will certainly occur.
This situation requires further reflection: Kolmogorov’s probability theory works
well as a mahematical theory, but the direct interpretation of its language leads to
counterintuitive results such as the one described above. An obvious solution is to
discard the the intuitive ideas “if P (A) = 0, then A is an impossible event” and “if
P (A) = 1, then A is a certain event”. Instead, we could say that “if P (A) = 0, then
A is a very unlikely event”, and “if P (A) = 1, then A is an almost certain event”, but
this solution has as a consequence that the correspondence between mathematical
formulas and their interpretation is now quite vague and far from intuition.
6.2 Non-Archimedean probability
There is a chance for everyone.
Helloween
Up to now, we have only identified some problems of Kolmogorov’s probability,
but we don’t have any ideas about how to overcome them. A useful insight will
come by studying the example of fair lotteries.
Definition 6.2.1 (Fair Lotteries). A fair lottery over Υ is a probability function P
that mirrors the intuitive properties we expect:
1. since every “ticket” must have a probability to be extracted, P must be defined
over all P(Υ);
89

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
2. the probability to extract every “ticket” is the same, so P ({υ}) does not depend
on the choiche of υ ∈ Υ;
3. we expect to calculate the probability of an arbitrary event by a process of
summing over the individual tickets: P (A) =
υ∈A P ({υ}).
Example 6.2.2 (Finite fair lotteries). Suppose that Υ = {1, . . . , n} ⊂ N. Then, a
fair lottery on Υ is described by a function P satisfying
and therefore
P ({υ}) = 1
|Υ|
P (A) = |A|
n
= 1
n
Let’s now focus on the fair lottery over N. If we want to follow an approach similar
to the one used in the finite case, we need to assume that the range of the probability
function has to include infinitesimals. Moreover, if we want to extrapolate the
intuitions concerning finite lotteries to infinite ones, we need a tool that allows to
generalize to the infinite case properties that hold for the finite case. In our theory,
we have already seen that, thanks to Transfer Principle, Ω-limit has such properties,
so we will use it in order to define the behaviour of Non-Archimedean Probability.
Now, we want to suggest a new approach at probability, that solves both problems
mentioned in Remark 6.1.5. The two main ideas are the following:
1. we use a non-Archimedean field as the range of probabilities: this allows to
make clear the ideas of “very unlikely” and “almost certain” events;
2. we use Ω-limit to give sense to probabilities of infinite sets. In particular, this
will ensure that the probability will be defined on all subsets of Ω and will
satisfy some good properties that hold in the finite case.
More formally, we will begin by giving new axioms for the Non-Archimedean Probability
within the context of Ω-Theory, and then we will show how those axioms
allow to solve the mentioned problems of Kolmogorov probability. In what follows,
we will always suppose Ω ⊇ Υ.
NAP 1 (Domain and Range). The events are all the events of P(Υ) and the probability
is a function
P : P(Υ) → R ∗
NAP 2 (Positivity). For all A ∈ P(Υ), P (A) ≥ 0.
NAP 3 (Normalization). For all A ∈ P(Υ),
P (A) = 1 ⇔ A = Υ
NAP 4 (Additivity). For all A, B ∈ P(Υ), if A ∩ B = ∅, then P (A ∪ B) =
P (A) + P (B).
90

6.3. FIRST CONSEQUENCES OF NAP AXIOMS
NAP 5 (Non-Archimedean Continuity). For all λ ∈ Pfin(Υ), λ = ∅, P (A|λ) ∈ R,
and
P (A) = lim
λ↑Υ P (A|λ)
Before we continue, we want to remark that the axioms of non-Archimedean
probability imply that P ({υ}) = 0 for all υ ∈ Υ. In fact, let’s suppose that this is
not the case, so that there is υ0 ∈ Υ such that P ({υ0}) = 0. NAP3 ensures that
P (Υ) = 1, NAP4 implies that P ({υ0}) + P ({υ0} c ) = 1, and we immediately deduce
that P ({υ0} c ) = 1, but this is in contradiction with NAP3. What is the meaning
of the condition P ({υ}) > 0 for all υ ∈ Υ? Let’s pause for a moment and think
about an example: suppose we want to calculate the probability to roll a specific
number on a six sided die. In this case, Υ = {1, 2, 3, 4, 5, 6}, as we already know that,
for instance, number 7 will never occurr. This simple example can be extended to
arbitrary probability. What we impose with NAP3 is that every elementary event
of Υ has a positive probability to occurr, and we choose to ignore all events we
already know as impossible. In this way, NAP allows to regain the intuition that,
if “P (A) = 0”, then A is truly an impossible event (and hence can be omitted from
the sample space).
This is but one of the differences between NAP and Kolmogorov’s probability.
If the reader is interest in a thorough comparison between Kolmogorov axioms and
NAP axioms, we remand to [12].
6.3 First consequences of NAP axioms
As a first consequence of NAP axioms, we want to show how a probability function
P over an arbitrary set Υ can be interpreted. The goal is to achieve a formula that
extends in some sense equation 6.1.
Given a Non-Archimedean probability function P : P(Υ) → R ∗ , we begin by
defining a function p : Υ → R ∗ as follows:
p(υ) = P ({υ})
then, we choose an arbitrary point υ0 ∈ Υ and define a weight function w : Υ → R ∗
in the following way:
w(υ) = p(υ)
p(υ0)
This weight function will allow us to get the desired interpretation of P . The first
step toward this result is to prove that the range of w is a subset of R:
Lemma 6.3.1. The function w takes values in R.
Proof. Take υ ∈ Υ arbitrarily and set r = P ({υ}|{υ, υ0}). By NAP 5, r ∈ R and
r < 1. By definition of conditional probability, we have also
r =
p(υ)
p(υ) + p(υ0)
= w(υ)
w(υ) + 1
91

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
so we conclude
w(υ) = r
1 − r
The previous Lemma allows to express the probability function P by using the
weight function w.
∈ R
Lemma 6.3.2. For all λ ∈ Pfin(Ω) such that λ ∩ Υ = ∅,
P (A|λ ∩ Υ) =
υ∈λ∩A w(υ)
υ∈λ∩Υ w(υ)
Proof. Notice that, by the finiteness of λ and by definition of w, we have
P (A|λ ∩ Υ) =
υ∈λ∩A p(υ)
υ∈λ∩Υ
p(υ) =
υ∈λ∩A p(υ0)w(υ)
υ∈λ∩Υ
p(υ0)w(υ) =
υ∈λ∩A w(υ)
υ∈λ∩Υ w(υ)
(6.4)
Now, we’d like to take the Ω-limit of equation
6.4 to get the desired result. The
only thing that’s left to determine is limλ↑Υ υ∈λ w(υ).
Lemma 6.3.3.
lim
λ↑Υ
υ∈λ
w(υ) = 1
p(υ0)
Proof. By Lemma 6.3.2 and the fact that w(υ0) = 1, we have
p(υ0) = lim P ({υ0}|λ) = lim
λ↑Υ λ↑Υ
w(υ0)χλ(υ0)
υ∈λ w(υ)
= limλ↑Υ χλ(υ0)
limλ↑Υ υ∈λ w(υ)
Since p(υ0) > 0 by NAP 1 and χλ(υ0) is eventually equal to 1 (take λ = {υ0}), we
conclude.
Putting everything together, we obtain the desired result:
Proposition 6.3.4.
P (A) = p(υ0) ·
w(υ) (6.5)
Proof. By Lemma 6.3.2, Lemma 6.3.3 and NAP 5, we have the following chain of
equalities:
92
P (A) = lim
P (A|λ) =
λ↑Υ limλ↑Υ
limλ↑Υ
υ∈A ◦
υ∈λ∩A w(υ)
υ∈λ w(υ) = p(υ0) ·
w(υ)
υ∈A ◦

6.4. FAIR LOTTERIES BY NAP
Remark 6.3.5. Equation 6.5 is an important result, since it generalizes equation
6.1 to the case of arbitrary sample spaces. If Υ is finite, then it’s rather easy to see
that NAP and Kolmogorov’s probability coincide. If Υ is infinite, notice that this is
the best result we can get, since the equation
P (A) =
p(υ)
υ∈A ◦
is not well defined, because it’s generally false that p(υ) ∈ R for all υ ∈ Υ, and the
sum over hyperextended sets is well defined only in this case.
The next Proposition replaces σ-additivity:
Proposition 6.3.6. Let {Ai}i∈I be pairwise disjoint sets and define A =
i∈I Ai.
Then
P (A) = lim P (Ai|λ)
i∈I ◦
λ↑Υ
i∈I◦ Proof. For all λ ∈ Pfin(Ω), we have
i∈I
P (Ai|λ) =
◦
υ∈λ∩Ai w(υ)
υ∈λ∩Υ w(υ)
=
Taking the Ω-limit, we conclude.
υ∈A w(υ)
υ∈λ∩Υ
w(υ) = P (A|λ)
Remark 6.3.7. The reader should be warned that Proposition 6.3.6 does not say
that P (A) =
i∈I◦ P (Ai). The equality stated by Proposition 6.3.6 is a rather
trivial result that actually doesn’t tell us more information than what we already
have.
Remark 6.3.8. Given any NAP P , we can define from it an Archimedean probability
Parch by the position
Parch(A) = sh(P (A))
This probability Parch is defined on all subsets of Ω and is always finitely additive.
Moreover, from case to case, it has some interesting properties: regarding fair
lotteries, we remand to the considerations expressed in Remark 25 of [12] .
6.4 Fair lotteries by NAP
Thanks to the results of the previous section, we can show some examples of NAP
at work. We begin by studying the fair lotteries over infinite sets.
Example 6.4.1 (Fair lottery on N). Since the lottery is fair, we have p(n) = p(m) =
p for all n, m ∈ N. In this case, the normalization axiom requires that
p = 1
n∈N ◦
93

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
It’s easy to see that
n∈N ◦
p = lim
λ↑N
and, taking the Ω-limit, we conclude
from which we deduce
p(n) = |λ ∩ N| · p
n∈λ
n(N) · p = 1
p(n) = 1
n(N)
for all n ∈ N. Since n(N) is an infinite number, this result is consistent with the
intuition that one ticket will be extracted, but every ticket has a very small probability
of being extracted. In the same spirit, we observe that
P (A) = n(A)
n(N)
(6.6)
so that the probability of the event A is a (rigorous and formal) generalization of the
intuitive concept “the number of the elementary events in A divided by the number
all the possible cases”.
Equation 6.6 shows us an important link between NAP and numerosity. In fact,
if we want the fair lottery on N to satisfy additional properties, such as the fact
that P (E) = P (O), then we need to introduce this fact as an opportune axiom
of numerosity. In turn, this axiom will correspond to a particular choiche of the
ultrafilter UΩ and to a particular model of Ω-Theory. This is the approach to NAP
followed in [12].
In our case, it turns out that we already have an axiom of numerosity that entails
that the fair lottery on N satisfies P (E) = P (O): it’s Numerosity Axiom 1. In fact,
when studying the model of Ω-Calculus, we have already seen in Example 1.12.4 that,
under Numerosity Axiom 1, we can deduce the equality P (E) = P (O) = 1/2α. We
signal that, assuming this axiom, the fair lottery on N also satisfies the asymptotic
limit property, for which we remand to [12].
Now, we will give an example about how NAP solves the problem of “almost
impossible” and “almost certain” events.
Example 6.4.2 (Fair lottery on N, continuation). Let A = E ∪ {1}. Clearly,
A ⊃ E, so we expect P (A) > P (E). Using the obvious decomposition of A, we
have immediately
P (A) = P (E) + 1
α
and the desired inequality holds. If we had chosen B = E ∪ {1, 3, . . . , 57}, we would
have found that
P (B) = P (E) + 29
> P (A) > P (E)
α
and this rigorously express not only the idea that B is more likely to happen than A,
but also allows to compute precisely how much more likely is B to happen.
94

6.4. FAIR LOTTERIES BY NAP
Using Numerosity Axiom 1, we can generalize the reasonings to fair lotteries over
other number sets.
Example 6.4.3 (Fair lottery on Z). Reasoning as in the previous examples, we have
that, for all n ∈ Z,
1
P (n) =
2α + 1
and
P (A) = n(A)
2α + 1
This time, if we compute P (2Z), the probability of extracting an even integer, we get
|2Z ∩ λn,θ| = |Z ∩ λn,θ| + 1
2
(the term +1 is due to the presence of 0). We deduce
so, taking the Ω-limit, we get
P (2Z ∩ λn,θ)
P (Z ∩ λn,θ) = |Z ∩ λn,θ| + 1
2|Z ∩ λn,θ|
P (2Z) = 1 1
+
2 4α + 2
1
=
2 +
1
2|Z ∩ λn,θ|
while, on the other hand, the probability of extracting an odd number is
P (Z \ 2Z) = 1 1
−
2 4α + 2
This difference is due to the presence of the zero. If we consider the set 2Z \ {0},
we get
P (2Z \ {0}) = P (2Z) − P ({0}) = 1 1 1
+ −
2 4α + 2 2α + 1
1 1
= −
2 4α + 2
as expected. We can conclude that Numerosity Axiom 1 entails that even numbers
are “exactly one more more” than odd numbers.
Example 6.4.4 (Fair lottery on Q). In the case of the fair lottery on Q, we can spend
some words regarding conditional probability. For instance, if [a0, b0)Q ⊂ [a1, b1)Q,
we would expect the conditional probability to satisfy the formula
This equation is equivalent to
P ([a0, b0)Q|[a1, b1)Q) = b0 − a0
b1 − a1
(6.7)
n([a, b)Q) = (b − a) · n([0, 1)Q) (6.8)
since, given this equality, we can easily prove equation 6.7: from
P ([a, b)Q) =
n([a, b)Q)
n(Q)
= (b − a) · n([0, 1)Q)
n(Q)
95

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
we deduce
P ([a0, b0)Q|[a1, b1)Q) = P ([a0, b0)Q)
P ([a1, b1)Q
To see that equation 6.8 holds, it’s sufficient to compute
lim |[a, b)Q ∩ λ|
λ↑Ω
= b0 − a0
b1 − a1
Thanks to Numerosity Axiom 1 and by assuming the model of Ω-Calculus given in
section 1.11, we have
lim |[a, b)Q ∩ λ| = lim |[a, b)Q ∩ λn|
λ↑Ω λn↑Q
and, reasoning as in Proposition 1.11.1, we obtain that, eventually,
|[a, b)Q ∩ λn| = (b − a) · n
From here, it’s easy to deduce equation 6.8.
For the fair lottery on R, we would like to ask a similar condition:
P ([a0, b0)|[a1, b1)) = b0 − a0
b1 − a1
but this is not possible. To see why, consider the intervals [0, 1) ⊂ [0, π): for those
intervals, we would have
P ([0, 1)|[0, π)) = 1
π
but we can prove that, for all fair lotteries, P (A|B) ∈ Q ∗ .
Proposition 6.4.5. If (Υ, P ) is a fair lottery and A, B ⊆ Υ, then P (A|B) ∈ Q ∗ .
Proof. By definition,
Since eventually
P (A|B) =
P (A ∩ B)
P (B)
P (A ∩ B ∩ λ)
= lim
λ↑Υ P (B ∩ λ)
|A ∩ B ∩ λ|
|B ∩ λ|
the thesis follows by transfer of membership.
∈ Q
For the fair lottery on R, the best we can do is require
P ([a0, b0)|[a1, b1)) ∼ b0 − a0
b1 − a1
|A ∩ B ∩ λ|
= lim
λ↑Υ |B ∩ λ|
and this is satisfied in our model of Ω-Theory thanks to Numerosity Axiom 1. It’s
also easy to see that the fair lotteries over N, Z, Q and R are all compatible, in
the sense that, if we call PN, PZ, PQ and PR the respective fair lotteries and S =
{N, Z, Q, R}, we have
PS1(A) = PS2(A|S1)
for all S1 ⊂ S2 and Si ∈ S.
96

6.5. INFINITE SEQUENCE OF COIN TOSSES BY NAP
6.5 Infinite sequence of coin tosses by NAP
In this section, we want to consider the modelization of an infinite sequence of coin
tosses with a fair coin. In the Kolmogorovian framework, the infinite sequence of
coin tosses is modeled by the triple (Υ, U, µ). The sample space Υ = {H, T } N is the
space of sequence with values in the set {H, T } (namely, heads and tails). If υ ∈ Υ,
we will denote υ = (υ1, . . . , υn, . . .).
The σ-algebra U is the σ-algebra generated by the “cylindrical sets”: given a nple
of indices (i1, . . . , in) and a n-ple (t1, . . . , tk) of elements in {H, T }, a cylindrical
set of condimension n is defined as follows:
C (i1,...,in)
(t1,...,tk)
= {υ ∈ Υ : υik = tk}
represents the event that the ik-th
coin toss gives tk as outcome for k = 1, . . . , n.
The probability measure on the generic cylindrical set is given by
µ
= 2 −n
(6.9)
From the probabilistic point of view, C (i1,...,in)
(t1,...,tk)
C (i1,...,in)
(t1,...,tk)
Thanks to Caratheodory’s Theorem, this measure can be extended in a unique way
to all U.
In this model, we can clealy see the problems with the Kolmogorovian approach
discussed in section 6.1:
• every event {υ} ∈ U has probability 0 to occurr, but the union of all these
“seemingly impossible” events has probability 1;
• if F is a finite set and {υ} ⊂ F , then the conditional probability µ({υ}|F ) is
not defined. On the other hand, we know that {υ}|F it’s not the empty event,
so the conditional probability makes sense and we expect it to be 1/|F |;
• there are subsets of Υ for which the probability is not defined (namely, the
non-measurable sets).
Now, we will show how the same situation can be modeled by non-Archimedean
probability. In particular, we want that the NAP P over Υ satisfies the following
assumptions:
1. if F ⊂ Υ is finite, then
2. P satisfies equation 6.9:
P (A|F ) =
P
C (i1,...,in)
(t1,...,tk)
|A ∩ F |
|F |
= 2 −n
(6.10)
Experimentally, we can only observe a finite number of outcomes: both cylindrical
events and finite conditional probability are based on a finite number of observations.
In some sense, the two assumptions above are the “experimental data” on
which we want to construct our model.
97

CHAPTER 6. AN APPLICATION: NON-ARCHIMEDEAN PROBABILITY
The first property implies that the probability is fair, so every infinite sequence
of coin tosses in Υ will have probability 1/n(Υ). Equation 6.10 is the counterpart
of equation 6.8 in the case of a fair lottery on R: equation 6.10 implies that for any
µ-measurable set E, we will have P (E) ∼ µ(E), in analogy to what implies equation
6.8 in the fair lottery on R.
To show that we can find a non-Archimedean probability measure P over P(Υ)
that satisfies conditions 1 and 2, we only need to show an appropriate model for
Ω-Theory in which equation 6.10 holds. This coincides with the choiche of an appropriate
ultrafilter UCT over Pfin(Ω).
In order to define the ultrafilter UCT , we need to introduce some notation. If
υ 1 = (υ 1 1, . . . , υ 1 n) ∈ {H, T } n is a finite string and υ 2 = (υ 2 1, . . . , υ 2 k , . . .) ∈ {H, T }N ,
then we define
υ 1 ⋆ υ 2 = (υ 1 1, . . . , υ 1 n, υ 2 1, . . . υ 2 k, . . .)
In other words, the sequence υ 1 ⋆ υ 2 is obtained by the sequence υ 1 followed by the
sequence υ2 .
N Now, if σ ∈ Pfin {H, T } and n ∈ N, we define
λn,σ = {υ 1 ⋆ υ 2 : υ 1 ∈ {H, T } n and υ 2 ∈ σ}
and
ΛCT = N
λn,σ : n ∈ N and σ ∈ Pfin {H, T }
Now, observe that ΛCT can be extended to a filter over Pfin(Ω), that in turn can be
extended to a nonprincipal ultrafilter UCT . Now, we only need to show that, in the
corresponding model of Ω-Theory, equation 6.10 holds.
Proposition 6.5.1. Equation 6.10 holds in the model of Ω-Theory obtained by the
ultrafilter UCT .
Proof. Consider the cylinder C (i1,...,in)
(t1,...,tk) and take N = max in. Then, for every σ ∈
N {H, T } , we have
Pfin
λN,σ ∩ C (i1,...,in)
(t1,...,tk) = {υ ∈ λN,σ : υik = tk}
Then
λN,σ ∩ C (i1,...,in)
(t1,...,tk) = |{υ ∈ λN,σ : υik = tk}| = |λN,σ|
2n Since
|λN,σ| = 2 N · |σ|
we conclude
λN,σ ∩ C (i1,...,in)
= 2 N−n · |σ|
Taking the Ω-limit, we obtain
n
C (i1,...,in)
(t1,...,tk)
(t1,...,tk)
= lim
λN,σ↑Υ 2N−n · |σ| = 2 −n · lim
λN,σ↑Υ 2N · |σ| = 2 −n · n(Υ)
Thanks to the fact that P is fair, we conclude
P C (i1,...,in)
n
(t1,...,tk) =
n(Υ)
as we wanted.
98
C (i1,...,in)
(t1,...,tk)
= 2 −n

Chapter 7
Further ideas
We have many intuitions in our
life, and the point is that many of
those intuitions are wrong. The
question is: are we going to test
those intuitions?
Dan Ariely
Due to the finiteness of this paper, we couldn’t develop properly all the ideas
that arose during the study of Ω-Theory. While we hope to deepen their
study in the future, we have decided to give a brief sketch of what we think
as the most interesting topics that could be studied within Ω-Theory.
This chapter will be somehow different from the rest of this paper: we will often
omit proofs and the reader won’t find interesting, polished results, but only rough
ideas and intuitions. We do believe that some of those ideas will yeld interesting
results, but the truth of this statement is still to be proved.
7.1 The Cartesian Product Axiom for Numerosity
Since we can assume that Ω ⊃ R N for all N ∈ N, it would be interesting to extend
the results given in Sections 1.6 and 3.4 to subsets of R N . With this goal in mind, we
should assume as axioms for numerosity the Numerosity Axiom 1 first introduced
in Section 1.3 and a “cartesian product axiom” stating that the numerosity of the
cartesian product A × B is the product of the numerosity of A by the numerosity of
B.
Numerosity Axiom 2 (Cartesian Product Axiom). For all A ⊆ R N , B ⊆ R M
satisfying A × B ⊆ Ω, n(A × B) = n(A) · n(B).
This new axiom allows the desired generalization, but has the major flaw that
it doesn’t help us overcome the difficulties we’ve encountered in Section 3.4. In
particular, it’s extremely difficulty to prove some results about countable additivity,
even if if we settle for infinitely closeness instead of equality.
99

CHAPTER 7. FURTHER IDEAS
Now, we will show that the axioms of Ω-Theory plus Numerosity Axioms 1 and
2 admit a model. We have already seen in Theorem 4.5.5 that a model of Ω-Theory
is obtained by the ultrapower of Ω with respect to a nonprincipal fine ultrafilter U.
To show a model in which Numerosity Axioms 1 and 2 hold, we need to impose
further conditions on the ultrafilter:
• If U ⊃ ΛR, then Numerosity Axiom 1 holds as a consequence of Theorem 1.11.4
and Proposition 4.5.4.
• In the same spirit, if U ⊃ ΛN R
Numerosity Axiom 2 holds.
for all N ∈ N, then we could show that also
Unfortunately, there is a problem with the last request since, if N = M, then
ΛN R and ΛMR don’t have the FIP. We circumvent this inconvenient by defining a new
family of sets ΛC in a way that the corresponding ultrafilter satisfies all axioms of
Ω-Theory plus Numerosity Axioms 1 and 2.
We start by considering the sets λn,θ ∈ ΛR, and then, for all N ∈ N, we pose
λn,θ,N = λn,θ ∪ λn,θ × λn,θ ∪ . . . ∪ λn,θ × . . . × λn,θ =
N times
With this notation, λn,θ = λn,θ,1, and we will often omit the third index when
referring to them. Now, we can set
ΛC = {λn,θ,N : n = m!, m ∈ N, θ ∈ Θ, N ∈ N}
It’s now easy to show that, if we take the Ω-limit with respect to λn,θ,N, then
Numerosity Axiom 2 holds:
Proposition 7.1.1. If A ⊆ RM M ′
and B ⊆ R , then
N
k=1
λ k n,θ
lim
λn,θ,N ↑Ω |(A × B) ∩ λn,θ,N| = lim
λn,θ,N ↑Ω |A ∩ λn,θ,N| · lim |B ∩ λn,θ,N|
λn,θ,N ↑Ω
Proof. If N ≥ M + M ′ , then
|(A × B) ∩ λn,θ,N| = |(A × B) ∩ λ
but, using the properties of cartesian products,
M+M ′
(A × B) ∩ λn,θ M+M ′
n,θ
= (A ∩ λ M M ′
n,θ) × (B ∩ λn,θ) At this point, by the finiteness of λn,θ, we easily conclude
On the other hand,
|(A ∩ λ M M ′
n,θ) × (B ∩ λ
n,θ)| = |A ∩ λ M M ′
n,θ| · |B ∩ λn,θ| |A ∩ λn,θ,N| = |A ∩ λ M M ′
n,θ| , |B ∩ λn,θ,N| = |B ∩ λn,θ| so, putting everything together, the thesis follows.
100
|

7.2. DIMENSION BY Ω-THEORY
We’ve seen that ΛC seems to be the right choice of sets if we want to satisfy all the
desired numerosity. Moreover, it’s quite easy to see that ΛC has the FIP: if we take
λn,θ,N and λn ′ ,θ ′ ,N ′ ∈ ΛC then, by definition, λn,θ,1 ⊆ λn,θ,N and λn ′ ,θ ′ ,1 ⊆ λn ′ ,θ ′ ,N ′,
and we already know that λn,θ,1 ∩ λn ′ ,θ ′ ,1 = ∅. We can then consider the filter
FΛC generated by ΛC and, by another application of Zorn’s Lemma, we obtain a
nonprincipal ultrafilter UC that extends FΛC . Now, we will show that the Numerosity
Axioms hold in the ultrapower of Ω modulo UC.
Theorem 7.1.2. Ω-Axioms 1, 2, 3, 4 and Numerosity Axioms 1 and 2 hold in the
ultrapower ΩUC .
Proof. The axioms of Ω-Theory hold by Theorem 4.5.5.
Numerosity Axiom 1 holds as a consequence of Theorem 1.11.4 and Proposition
4.5.4.
Numerosity Axiom 2 holds as a consequence of Proposition 7.1.1.
7.2 Dimension by Ω-Theory
Thanks to the Cartesian Product Axiom, we can generalize some results of Section
1.6 and of Section 3.4 to subsets of R N . Since we are interested in the development
of a concept of measure for subsets of R N , we begin by showing explicitely the
generalization of Proposition 1.6.7 to arbitrary dimension.
Proposition 7.2.1. Let E = N n=1 [an, bn), with an and bn ∈ R. Then Numerosity
Axiom 2 implies
N
n(E)
∼ (bn − an)
n([0, 1)) N
Proof. By Numerosity Axiom 2,
n(E)
=
n([0, 1)) N
n=1
N
n=1
At this point, we conclude by Proposition 1.6.7.
n([an, bn))
n([0, 1))
Given a set E ⊆ R N , the ratio n(E) · n([0, 1)) −N plays an important role in the
development of a concept of dimension. For this reason, we find useful to fix it in a
definition.
Definition 7.2.2. If E ⊆ R N and r ∈ R + , we will call the normalized numerosity
of order r of the set E the ratio
n(E)
n([0, 1)) r
and we will denote it by nnr(E). When there will be no confusion about r, we will
often write nn(E) instead of nnr(E).
As an example of the use of normalized numerosity, we will rephrase some known
results by using this concept.
101

CHAPTER 7. FURTHER IDEAS
Example 7.2.3. Notice that, if r = 1, then the normalized numerosity of order 1
coincides with the notion given in Definition 3.2.3.
Now, suppose that a, b ∈ Q. We already know that, thanks to Numerosity Axiom
1, nn1([a, b)) = b − a. It’s also clear that, if r > 1, then nnr([a, b)) ∼ 0, since
nnr([a, b)) = nn1([a, b)) ·
1
1
= (b − a) ·
n([0, 1)) r n([0, 1)) r−1
and n([0, 1)) −r+1 is infinitesimal for all r > 1.
If we consider the set E = [a, b) × [c, d) ⊂ R 2 , then Numerosity Axiom 2 entails
nn2(E) ∼ (b − a) · (d − c). In the same spirit as the previous example, it’s also
easy to see that nn1(E) = nn2(E) · n([0, 1)) is an infinite number, while nnr(E) is
infinitesimal for all r > 2.
The definition of the normalized numerosity of order r and the ideas shown in
Example 7.2.3 bear some similarities with the definition of Hausdorff measure and
some of its consequences. Moreover, the same results show that, in analogy to the
definition of Hausdorff dimension from Hausdorff measure, it could be interesting to
give a notion of dimension based on normalized numerosity. Since we want to give
this definition for subsets of R N , we will use the setting of Ω-Theory plus Numerosity
Axioms 1 and 2. We also assume that the reader is familiar with Hausdorff measure
and dimension. For an introduction about those topics, we remand to [7].
We begin by defining the dimension of a set in the following way:
Definition 7.2.4. If E ⊆ R N , we set
dim(E) = inf{r ∈ R + : nnr(E) ∼ 0}
Given this definition, we can generalize and make precise the results found in
Example 7.2.3:
Proposition 7.2.5. If E = N
n=1 [an, bn) and we don’t exclude that an = bn for some
n ≤ N, then
dim(E) ≤ N
More precisely, if we call A = {n ≤ N : an = bn}, we have
Proof. By Numerosity Axiom 2, we have
hence, for all r ∈ R + ,
n(E) =
dim(E) = |A|
N
n([an, bn)) =
n([an, bn))
n=1
n∈A
n∈A
nnr(E) =
1
nn1([an, bn)) ·
n([0, 1)) r−|A|
Observe that, by hypothesis,
n∈A nn1([an, bn)) is a finite, positive number. We
immediately deduce nnr(E) ∼ 0 for all r > |A| and nn|A|(E) =
n∈A nn1([an, bn)) >
0, as we wanted.
102

7.3. NUMEROSITY AND LEBESGUE MEASURE
The next step would be to generalize those ideas to arbitrary subsets of RN .
Unfortunately, there is a problem with subsets whose normalized numerosity is an
infinite number. As an interesting and fundamental example, take the case of the
set R: the normalized numerosity of order 1 of R is an infinite number and, when
r > 1, we don’t know how to extimate the ratio n(R) · n([0, 1)) −r . This is essential
in order to proceed since, if we find that there exists some positive s ∈ R satisfying
n(R)
sh
n([0, 1)) s
< +∞ (7.1)
then the dimension of R would be s. Unfortunately, it’s not straightforward to
determine if such a s exists and, even if the existence of s is settled, one needs also
to calculate its exact value. Moreover, there is the possibility that the validity of
equation 7.1 depends upon the choiche of the model of Ω-Theory, and so a new
axiom could be necessary to rule the dimension of R.
Then, it would naturally arise also the question if there could be some mathematical
interest in giving a definition of dimension where the range of all possible
dimension is all R ∗ . In this case, we should find out what could be a good definition,
and then study its consequences and possible applications.
7.3 Numerosity and Lebesgue measure
I teoremi non si dimostrano a
comando.
Gianni Gilardi
During the study of Ω-Theory, we have spent quite some time in the attempt
to find some axioms of numerosity that would entail the equivalence between numerosity
and Lebesgue measure (and, as a consequence, between Ω-integral and
Lebesgue integral). At first, we tried in many ways to give a countable additivity
result for numerosity assuming only the validity of Numerosity Axiom 1, without
success. After those failures, we tried to impose the desired result by a new axiom
of numerosity: it turned out that one of such axioms exists, but it isn’t powerful
enough to entail the equivalence of numerosity and Lebesgue measure. In particular,
this axiom didn’t entail the equality sh(nn1([0, 1)Q)) = 0, against the well known fact
that µ([0, 1)Q) = 0.
Our more recent attempt is to prove the coherence with Ω-Theory of the following
axioms:
1. Numerosity Axiom 1 and
2. if E and F are two Lebesgue measurable sets satisfying µ(E) < µ(F ), then
also nn1(E) < nn1(F ).
By a simple argument, we can easily show that the validity of both requests would
immediately imply that nn1(E) ∼ µ(E) . In fact, for all q ∈ Q satisfying µ(E) < q,
103

CHAPTER 7. FURTHER IDEAS
we can argue that nn1(E) < nn1 [0, q) = q, and the same idea can be used to show
the validity of the other inequality.
Unfortunately, it’s still an open question wether those two axioms are compatible
with Ω-Theory. If this turns out to be the case, then also the desired equivalence
between the Ω-integral and Lebesgue integral would easily follow from the result
shown above. On the other hand, if the two axioms are not compatible with Ω-
Theory, it would be a very interesting research field to determine if there can exists
an axiom that entails the equivalence between numerosity and Lebesgue measure,
or if such equivalence is incompatible with the theory.
We signal that another possible route toward a result of equivalence between
numerosity and Lebesgue measure is given by the natural extension of numerosity,
a topic that will be briefly presented in the last section of this chapter.
7.4 Functional Analysis by Ω-Theory
We want to spend a few words about how the notions of functional analysis can be
given within Ω-Theory. Since we know that, by Transfer Principle, the notions of
R-vector space, metric, seminorm, norm, bilinear form and scalar product can be
extended by using Ω-limit in a way that their elementary properties are preserved,
we could easily give the definitions of R ∗ -vector spaces and of metrics, seminorms,
norms, bilinear forms and scalar products that take values in R ∗ and study their
behaviour, with some applications of functional analysis in mind. Moreover, the
definition of uniform topology, introduced in the real case by Definition 2.2.1, can
be extended to arbitrary sets, and we believe that many other notions of topology
and differential geometry can be given in the same way. It could be an interesting
research topic to study if those notions together with Ω-limit could be used to obtain
some interesting results of functional analysis, topology and differential geometry.
7.5 R ∗∗
Given the axioms of Ω-Theory, one can wonder wether the Ω-limit of functions
ϕ : Fun(Pfin(Ω), R∗ ) can be defined. The answer is yes, and we can indeed develop
a theory for R∗∗ = (R∗ ) ∗ = limλ↑Ω cR∗(λ). Since a thorough study of this topic runs
beyond the scope of this paper, in this section we will only give a few examples of
the behaviour of Ω-limit of hyperreal numbers. For this reason, the main “inference
tool” we will use throughout the section will be Transfer Principle since, if used
correctly, it allows to easily deduce some meaningful inequalities.
We find interesting to begin the studt of R∗∗ by showing some examples of the
behaviour of Ω-limit of constant functions.
Example 7.5.1. Let’s pick ξ ∈ R ∗ , ξ ∼ 0 and consider the function cξ(λ). By
definition of infinitesimal number,
104
cξ(λ) < 1
n
for all n ∈ N

and, by Transfer of inequalities, we deduce
lim cξ(λ) <
λ↑Ω 1
ν
for all ν ∈ N∗
7.5. R ∗∗
By Lemma 5.2.6, we conclude immediately that limλ↑Ω cξ(λ) = ξ.
In the same spirit, if ξ is infinite, then limλ↑Ω cξ(λ) > ν for all ν ∈ N ∗ and, with
a similar argument, we conclude that also limλ↑Ω cξ(λ) = ξ.
If ξ is finite, then ξ = sh(ξ) + ɛ, where ɛ is infinitesimal, hence
lim cξ(λ) = sh(ξ) + lim cɛ(λ)
λ↑Ω λ↑Ω
and, by the first part of this example, we deduce limλ↑Ω cξ(λ) ∼ ξ.
Motivated by this example, we will introduce a new definition, that will be a
useful short-hand.
Definition 7.5.2. For all ξ ∈ R ∗ , we will call ξ ∗ = limλ↑Ω cξ(λ).
The reasonings shown in the previous example should be sufficient to convince
the reader of the validity of the following statements:
• ξ ∗ = ξ if and only if ξ ∈ R;
• sh(ξ) = sh(ξ ∗ ).
We can now generalize the ideas of Example 7.5.1 to arbitrary functions.
Example 7.5.3. Consider the function ϕ(λ) = |λ|/n(R). Clearly, this is a strictly
increasing function in the sense that, if λ2 ⊃ λ1, then ϕ(λ2) > ϕ(λ1). The intuition
could suggest that the Ω-limit of this function must be 1 but, by using correctly
transfer principle, we deduce that
lim ϕ(λ) =
λ↑Ω n(R)
n(R) ∗
Reasoning as in Example 7.5.1, we also deduce
n(R) 1
<
n(R) ∗ ν
for all ν ∈ N∗
In the same way, one can prove that, if ϕ(λ) is infinitesimal a.e., then
lim ϕ(λ) <
λ↑Ω 1
ν
for all ν ∈ N∗
This property is somehow in contrast with the intuition that some infinitesimal functions
could have finite Ω-limit.
105

CHAPTER 7. FURTHER IDEAS
Example 7.5.4. Consider the sequence of hyperreal numbers {ξn}n∈N, and suppose
that ξn is finite for all n ∈ N. We could then define
lim ξn ∈ R
λ↑N
∗∗
n∈λ
and wonder if this definition has some good properties. For every λ ∈ Pfin(Ω), we
can write
ξn =
sh(ξn) + ɛλ
n∈λ∩N
n∈λ
n∈λ∩N
where ɛλ is infinitesimal. Taking the Ω-limit, we obtain
lim ξn = lim sh(ξn) + lim ɛλ
λ↑N
λ↑N
λ↑Ω
and, by Example 7.5.3, we already know that limλ↑Ω ɛλ ∼ 0, so we can conclude
lim ξn ∼
λ↑N
sh(ξn)
n∈λ
n∈λ
n∈N ◦
7.6 Ω-integral of hyperreal functions
Once we have defined the field R ∗∗ , one of the first temptations is to extend the
definition of Ω-integral to functions with hyperreal range. More precisely, we can
take Definition 3.2.1 and simply extend the class of functions for which the Ω-integral
is defined. As a result, we would get the following definition:
Definition 7.6.1. For any f : E ⊆ R → R∗ we pose
f ◦ 1
(x)dΩx =
n([0, 1)) lim
f(x)
λ↑E
E ◦
Unfortunately, the integral defined above isn’t linear over R ∗ . In fact, it’s easy
to show that linearity is replaced by another property:
Proposition 7.6.2. According to Definition 7.6.1, if f : E → R and ξ ∈ R∗ , then
ξ · f ◦ (x)dΩx = ξ ∗
· f ◦ (x)dΩx
E ◦
Proof. By the properties of Ω-limit, we have the following chain of equalities:
ξ · f ◦ (x)dΩx =
1
n([0, 1)) lim
ξ ·
λ↑E
f(x)
E ◦
so the thesis follows.
106
E ◦
x∈λ
x∈λ
=
ξ∗ n([0, 1)) lim
f(x)
λ↑E
x∈λ
= ξ ∗
· f ◦ (x)dΩx
E ◦

7.7. THE DIRAC DISTRIBUTION
In order to avoid this problem, a completely new definition is necessary. First of
all notice that, if f : R → R ∗ then, for every x ∈ R, we can find ϕx(λ) : Pfin(Ω) → R
satisfying limλ↑Ω ϕx(λ) = f(x). Moreover, if f(x) ∈ R we can suppose that ϕx(λ) =
cf(x)(λ). Since f(x) = limλ↑Ω ϕx(λ) for all x ∈ R, we can define the Ω-integral of f
in the following way:
Definition 7.6.3. For any f : E ⊆ R → R∗ , we pose
f ◦ 1
(x)dΩx =
n([0, 1)) lim
λ↑Ω
E ◦
x∈λ∩E
As we did before, we will usually denote this integral by
We remark that, according to Definition 7.6.3,
f(x)dΩx ∈ R ∗
E ◦
ϕx(λ)
E ◦ f(x)dΩx.
so we discovered that the field R ∗∗ isn’t really necessary to define the Ω-integral of
functions that take value in R ∗ .
It’s easily verified that this integral is independent on the choiche of the functions
ϕx and, if f : R → R, then Definition 3.2.1 and Definition 7.6.3 give rise to the same
notion of Ω-integral, since it’s sufficient to take ϕx(λ) = f(x) for all λ ∈ Pfin(Ω)
and for all x ∈ R. Moreover, it’s rather easy to see that linearity over R ∗ holds.
Proposition 7.6.4. According to Definition 7.6.3, if f : E → R∗ and ξ ∈ R∗ , then
ξ · f(x)dΩx = ξ · f(x)dΩx
E ◦
Proof. Pick a function ψ : Pfin(Ω) → R satisfying limλ↑Ω ψ(λ) = ξ. Then, we have
the following chain of equalities
as we wanted.
E ◦
ξ · f(x)dΩx =
=
1
n([0, 1)) lim
λ↑Ω
1
n([0, 1))
E ◦
x∈λ∩E
lim ψ(λ) · lim
λ↑Ω λ↑E
=
ξ
n([0, 1)) lim
ϕx(λ)
λ↑E
x∈λ
= ξ · f(x)dΩx
7.7 The Dirac distribution
E ◦
(ψ(λ) · ϕx(λ))
ϕx(λ)
In this section, we want to show how the well known distribution of the Dirac mass
can be defined by Ω-Theory. We suppose that the reader is familiar with the topic
x∈λ
107

CHAPTER 7. FURTHER IDEAS
and we remand to Example 4.1 of Chapter 3 of [4] for a reference. We also signal that
we will oversimplify the presentation of this topic, and we will omit some important
details and definitions in order to focus on the original approach given by Ω-Theory.
We begin by recalling briefly the definition of the Dirac mass.
Definition 7.7.1. For a matter of simplicity, suppose that K ⊂ R is a compact
set and v ∈ C∞ (K). Given x0 ∈ K, we define the Dirac mass centered at x0 the
distribution satisfying
δx0v = v(x0) (7.2)
where the integral is intended in the sense of distributions.
K
In the context of Ω-Theory, we are tempted to define a function ∂x0 : R → R∗ such that
K◦ (∂x0 · v)(x)dΩx = v(x0)
Using the Ω-integral given by Definition 7.6.3, it’s immediate to see that, if we pose
∂x0 = n([0, 1)) · χ{x0}, then
K◦ (∂x0 · v)(x)dΩx =
n([0, 1)) · v(x0)
n([0, 1))
= v(x0)
It seems to us that this approach shows a number of interesting attributes. First
of all, we find very appealing that the Dirac mass can be defined as a function
whose Ω-integral yelds the desired result. This is in open contrast with the classical
theory, where one needs to extend the meaning of the integral to take into account
the fact that a function that is equal to 0 almost everywhere with respect to µ has
nevertheless a nonzero integral. Moreover, we think that the definition of the Dirac
mass in the context of Ω-Theory is simpler and entails in an intuitive way the sense
of the distribution that, in the classical theory, has to be retraced in the alternate
definition of δx0 as limit of simple functions.
It’s still an open question if this approach can be extended with similar benefits
to the whole theory of distributions, or if the conceptual simplification happens only
in the case of the Dirac mass.
7.8 The natural extension of numerosity
In the setting of Ω-Theory and without assuming any axiom of numerosity, we’ve
seen in Section 1.6 that numerosity is a function n : P(Ω) → N ∗ that’s always
monotone and finitely additive. By Transfer Principle, we conclude that the natural
extension of numerosity is monotone and hyperfinitely additive in a sense that can
be precised. It’s only natural to wonder if this new function can replace numerosity
as basic tool for “counting” the number of elements of sets.
For a matter of commodity, we will start by focus on the restriction of n to P(R):
from the definition
n : P(R) → N ∗
108

we obtain that
7.8. THE NATURAL EXTENSION OF NUMEROSITY
n ∗ : P(R) ∗ → N ∗∗
Explicitely, n∗ is defined on the set
P(R) ∗
= lim ψ(λ) : ψ(λ) ∈ P(R) for all λ ∈ Pfin(Ω)
λ↑Ω
From this definition we can already deduce some properties of n ∗ .
Proposition 7.8.1. For all A ⊆ R, n ∗ (A ∗ ) = (n(A)) ∗ .
Proof. The proof is almost trivial: the following chain of inequalities hold by definition
n ∗ (A ∗ ) = lim n(cA(λ)) = lim cn(A)(λ) = n(A)
λ↑Ω λ↑Ω ∗
so the thesis follows.
With a similar argument, we can show that Numerosity Axiom 1 implies that a
similar property holds for n ∗ and for all ξ, ζ ∈ Q ∗ . We begin by showing that, for
all a, b ∈ Q, then the ratio n ∗ ([a, b) ∗ ) · n ∗ ([0, 1) ∗ ) assumes the expected value.
Proposition 7.8.2. By assuming Numerosity Axiom 1, for all a, b ∈ Q,
n ∗ ([a, b) ∗ )
n ∗ ([0, 1) ∗ )
= b − a
Proof. With an argument similar to the proof of Proposition 7.8.1,
n ∗ ([a, b) ∗ )
n ∗ ([0, 1) ∗ )
n(c[a,b)(λ))
= lim
λ↑Ω n(c[0,1)(λ))
= lim(b
− a) = b − a
λ↑Ω
In the same spirit, we can easily extend the equality of Proposition 7.8.2 to all
ξ, ζ ∈ Q ∗ .
Proposition 7.8.3. By assuming Numerosity Axiom 1, for all ξ, ζ ∈ Q∗ , if ξ =
limλ↑Ω ϕ(λ) and ζ = limλ↑Ω ψ(λ) and by calling [ξ, ζ)R∗ = limλ↑Ω [ϕ(λ), ψ(λ)), we
have
n∗ ([ξ, ζ)R∗) n∗ ([0, 1) ∗ = ξ − ζ
)
Proof. By definition,
n ∗ ([ξ, ζ)R ∗)
n ∗ ([0, 1) ∗ )
n([ϕ(λ), ψ(λ)))
= lim
λ↑Ω n([0, 1))
= lim
λ↑Ω ϕ(λ) − lim
λ↑Ω ψ(λ) = ξ − ζ
Corollary 7.8.4. By assuming Numerosity Axiom 1, for all ξ, ζ ∈ R∗ , if ξ =
limλ↑Ω ϕ(λ) and ζ = limλ↑Ω ψ(λ) and by calling [ξ, ζ)R∗ = limλ↑Ω [ϕ(λ), ψ(λ)), the
following equality holds
n∗ ([ξ, ζ)R∗) n∗ ([0, 1) ∗ ∼ ξ − ζ
)
109

CHAPTER 7. FURTHER IDEAS
Proof. This result follows from Proposition 7.8.3 and by the fact that the Ω-limit of
functions that take only infinitesimal values is infinitesimal.
Now, we will make precise the statement that n∗ is hyperfinitely additive. If
{An}n∈N are pairwise disjointed subsets of R and µ = limλ↑Ω ϕ(λ) ∈ N∗ , then the
following equality holds by definition:
n ∗
⎛ ⎛ ⎞⎞
ϕ(λ)
⎝lim ⎝
ϕ(λ)
⎠⎠
= lim n(An) (7.3)
λ↑Ω
n=1
An
λ↑Ω
n=1
∗ Now, it would be nice to relate the first member of equation 7.3 to n n∈N An
∗ .
From Proposition 1.7.11, we know that
⎛
ϕ(λ)
lim ⎝
⎞
⎠ ⊆
∗ (7.4)
λ↑Ω
n=1
An
and the equality holds only if the union is finite. Since we are interested in the case
where the union is infinite, we already know that the two sets can’t be equal.
It seems that we are facing the same difficulties we encountered when studying
numerosity: even if, in this case, we have equation 7.3, we don’t know if it’s of
any help. In fact, in the next Example, we will show that this impression is wrong
and, by assuming Numerosity Axiom 1, we can already use equation 7.3 to get some
interesting results.
Example 7.8.5. Consider the sets {[1/2 n+1 , 1/2 n )}n≥0. It’s clear that
n≥0
n∈N
An
[1/2 n+1 , 1/2 n ) = (0, 1)
Now, pick µ = max(N◦ ) and, thanks to equation 7.3, consider the equality chain
n∗ max(λ) limλ↑N n=0 [1/2n+1 , 1/2n
)
n∗ ([0, 1) ∗ max(λ) 1 1
= lim = 1 − ∼ 1
)
λ↑N 2n+1 2 µ+1
n=0
On the other hand, in this case,
from which we deduce
(0, 1) ∗ \
(0, 1) \
⎛
⎝lim
λ↑N
k
n=0
[1/2 n+1 , 1/2 n ) = (0, 1/2 n+1 )
max(λ)
[1/2 n+1 , 1/2 n ) ⎠ = (0, 1/2 µ+1 )R∗ n=0
and, thanks to Corollary 7.8.4, we already knew that
110
n ∗ ((0, 1/2 µ+1 )R ∗)
n ∗ ([0, 1) ∗ )
⎞
∼ 1/2 µ+1 ∼ 0

7.8. THE NATURAL EXTENSION OF NUMEROSITY
We do believe that the ideas used in the previous example could be extended to
other, more general cases without much effort. We want to explicitely observe that,
thanks to Proposition 7.8.1, some results about natural extension of numerosity can
be used to infer similar properties of numerosity, and this applies in particular for
any result about countable additivity.
It’s our hope to deepen the study of this topic in the near future.
111

BIBLIOGRAPHY
Bibliography
[1] Vieri Benci, Mauro Di Nasso, Alpha Theory: Mathematics with Infinite and
Infinitesimal Numbers (paper in preparation, 2012)
[2] Vieri Benci, Mauro Di Nasso, Alpha Theory: An Elementary Axiomatics for
Nonstandard Analysis (Expositiones Mathematicae, 21-2003)
[3] Gianni Gilardi, Analisi Matematica di Base (McGraw-Hill, 2001)
[4] Gianni Gilardi, Analisi Tre (McGraw-Hill, 1994)
[5] Gianni Gilardi, Analisi Funzionale, available for free download at http://wwwdimat.unipv.it/gilardi/WEBGG/PSPDF/analfunz1011.pdf
[6] Haim Brezis, Carlo Sbordone, Analisi Funzionale, Teoria ed applicazioni - Con
un’appendice su Integrazione Astratta di Carlo Sbordone (Liguori Editore, 1986)
[7] Gianni Gilardi, Misure di Hausdorff e applicazioni, available for free download
at http://www-dimat.unipv.it/gilardi/WEBGG/PSPDF/hausd.pdf
[8] I. N. Herstein, Algebra (Editori Riuniti, Roma 1994)
[9] R. Schoof, B. van Geemen, Algebra, available for free download at http://wwwdimat.unipv.it/canonaco/notealgebra.pdf
[10] Andrea Cantini, Pierluigi Minari, Introduzione alla logica - parte
II, available under the authorization of the authors at the page
http://www.philos.unifi.it/CMpro-v-p-60.html
[11] Dirk Van Dalen, Logic and Structure (Springer, 2008)
[12] Vieri Benci, Leon Horsten, Sylvia Wenmackers, Non-Archimedean Probability,
available for free download at the page http://arxiv.org/abs/1106.1524
[13] Appunti delle lezioni di Probabilità e Statistica tenute dal professor
Eugenio Regazzini, available for free download at http://wwwdimat.unipv.it/∼bassetti/didattica/probI/probabilitaNEW.pdf
[14] Jean Jacod, Philip Protter, Probability Essentials (Springer, 2003)
[15] Abraham Robinson, Non-Standard Analysis (North-Holland, 1966)
112

BIBLIOGRAPHY
[16] Imre Lakatos, Proofs and Refutations - The logic of mathematical discovery
(Cambridge University Press, 1976)
[17] Roberta De Giambattista, Concetti e definizioni dell’analisi matematica dal
punto di vista didattico (Thesis, 2010)
[18] Dan Ginsburg, Brian Groose, Josh Taylor, Bogdan Vernescu, The History of
the Calculus and the Development of Computer Algebra Systems, available for
free consultation at http://www.math.wpi.edu/IQP/BVCalcHist/index.html
113

ACKNOWLEDGMENTS AND THANKS
Acknowledgments and Thanks
What I’m trying to say is that
there is always something that
makes your day. Just look. And if
you don’t find it, look again. Look
closer.
Iris B.
La possibilità di studiare matematica è stata una delle tre occasioni più
grandi che io abbia mai avuto. Se sono riuscito ad arrivare fino a qui, il
merito non è solamente mio, ma anche di tutte le persone che, in un modo
o nell’altro, mi hanno insegnato ed aiutato a crescere e ad affrontare le sfide di tutti
i giorni. A ciascuno di loro, dedico qualche parola di ricordo e ringraziamento.
Acknowledgment 1. To Eleonora.
Like an autumn day, you are wonderful and fascinating.
Like a star, you are bright and splendid.
Like a flame that never dies, you are warm and welcoming.
Like π, you are real, and yet transcendental.
Fortunately, the proof of those statements runs way beyond the scope of this paper.
Ringraziamento 2. Questa tesi non sarebbe mai stata realizzata senza il supporto
costante e significativo della mia famiglia. Sono molto grato ai miei genitori per
avermi appoggiato in ogni momento della mia vita universitaria e non, agevolandomi
in ogni necessità, compresa quella di visitare periodicamente Pisa per poter studiare
l’affascinante argomento presentato in questa tesi. Vi ringrazio anche per la fiducia
che mi avete dimostrato in questi anni, che si è sempre manifestata nelle piccole e
nelle grandi cose.
Ringrazio moltissimo anche la Zia, soprattutto per la sua disponibilità di condividere
insieme alcuni momenti impegnativi (come andare a comperare i vestiti) e
di riflessione, e per il suo atteggiamento positivo nei confronti del mondo. La tua
presenza serena e il tuo buon senso sono ammirevoli, soprattutto per me che sono
carente di certe qualità.
Ringrazio anche mio fratello Angelo e mia sorella Paola, per tutte le occasioni
di crescita che abbiamo vissuto insieme. Un pensiero particolare è riservato anche a
Marie, che ha agevolato molte di queste occasioni. Mes meilleurs souhaits pour un
avenir heureux!
114

RINGRAZIAMENTI
Ricordo con riconoscenza e gratitudine anche la famiglia Tiengo che, in tutti
questi anni, si è sempre dimostrata molto accogliente e disponibile nei miei confronti.
Vi sono molto grato per tutto quello che avete fatto per me, e per tutte le attenzioni
e le gentilezze che mi avete sempre riservato.
Ringraziamento 3. Il momento è quanto mai opportuno per ricordare e ringraziare
tutti gli amici presenti e passati. Tra questi, un posto di riguardo è dedicato
a Gianluca: la nostra lunga e sfaccettata amicizia è un tesoro da custodire e da
coltivare, a qualsiasi costo. Un ringraziamento speciale anche a tutta la famiglia
Lanati, per avermi sempre accolto come loro “secondo figlio”.
Un ringraziamento particolare è riservato anche ad Andrea, con il quale ho sempre
condiviso degli ottimi momenti di svago e di riflessione. Mi auguro che possano
ripetersi a lungo nel nostro futuro.
Ricordo con piacere anche Maria Grazia Toso e il nostro complesso rapporto
tra “creativo” e “ricettivo” (Ch’ien e K’un), la cui ultima espressione si trova nella
dimostrazione di un Lemma di questa tesi.
Ringrazio anche Giulia, Caterina, Dario Merzi, Dario Mazzoleni, i miei compagni
di corso, tutti gli amici del SELP e della Scuola Estiva di Logica, Giovanna, Lorenzo,
Giulio, Nello, Lame, Johnny, Mich, Luca, Andrea Colonna, Florenc e tutti coloro
che in qualche modo hanno lasciato un segno positivo nella mia vita.
Ringraziamento 4. Il mio interesse nei confronti dell’Alpha-calcolo e dell’Alphateoria
è nato durante l’edizione del 2010 della Scuola Estiva di Logica organizzata
dall’AILA. Sono molto grato al professor Pierluigi Minari per avermi dato la possibilità
di conoscere questo evento e a Caterina per averlo condiviso insieme a tante
altre occasioni di studio.
La mia gratitudine va anche ai professori Vieri Benci e Mauro Di Nasso, per
avermi reso partecipe del loro lavoro e per avermi accompagnato con grandissima
disponibilità e cura durante lo studio di questo argomento così affascinante. Ringrazio
molto anche il professor Vitali che, nonostante gli impegni, ha seguito con
impegno, passione e pazienza l’evoluzione di questa tesi. Gli sono particolarmente
riconoscente per i suoi preziosi spunti di riflessione, che hanno sempre portato buoni
frutti.
Sono molto grato al SELP, a Giorgio Venturi e a Samuele Maschio per avermi
dato la possibilità di raccontare le basi di questa tesi durante un seminario SELP a
Pavia nel maggio 2011.
Ringraziamento 5. Ringrazio in modo particolare il professor Gilardi, il professor
Boffi, la professoressa Reggiani, il professor Antonini e tutti gli altri professori che
ho incontrato durante i miei studi universitari. Vi sono molto grato per la vostra
grandissima disponibilità e per la passione e competenza con cui vi dedicate all’insegnamento.
Grazie a voi, durante i miei studi universitari sono cresciuto non solo
nella mia matematica.
Ricordo anche tutte le maestre e le professoresse di matematica di cui sono stato
alunno, ed in particolare la maestra Lucia, la maestra Elena, la prof. Ghini, la
prof. Bianchi e la prof. Cazzola. Ringrazio anche la professoressa Sartirana e la
professoressa Cova per avermi guidato, valorizzato e fatto crescere nel mio tutt’ora
prezioso rapporto con la letteratura.
115

ACKNOWLEDGMENTS AND THANKS
Ringraziamento 6. Infine, un pensiero di gratitudine e riconoscenza è sempre rivolto
al Professor Marni, al Dottor Carlo Campana, al Dottor Ghio, a Milena e
a tutte le infermiere e specializzande dell’ambulatorio scompensi e trapianti della
Cardiologia, al Professor Mario Viganò, al Dottor Carlo Pellegrini, a tutta la Cardiochirurgia,
alla Dottoressa Barbara Petracci, alla Dottoressa Angela Di Matteo e
a tutte le persone che ho dimenticato ma che mi hanno permesso di essere qui dopo
così tanto tempo e dopo così tante avventure.
116
06-06-2001 − 07-02-2012
Ten thousand miles are quite long. . .
. . . but not too far!