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

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
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