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

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
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(ϕ + ψ)(a) = ϕ(a) + ψ(a)
(ϕ · ψ)(a) = ϕ(a) · ψ(a)