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

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:
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1. E ∗ ∩ F ∗ = (E ∩ F ) ∗ ;
2. E ⊆ F implies E ∗ ⊆ F ∗ ;