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

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
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