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

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