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

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CHAPTER 1. Ω-CALCULUS Proposition 1.6.9. Let f : R → R satisfy |f(F )| = |F | for all finite subsets of R. If for all E ⊆ R the set {λ ∈ Pfin(Ω) : |f(E ∩ λ)| = |f(E) ∩ λ|} is qualified, then n is invariant with respect to f. Proof. By hypothesis, |E ∩ λ| = |f(E ∩ λ)| and, since |f(E ∩ λ)| = |f(E) ∩ λ| almost everywhere, by transfer of equality we conclude Hence the thesis. lim |f(E) ∩ λ| = lim |f(E ∩ λ)| = lim |E ∩ λ| λ↑Ω λ↑Ω λ↑Ω It’s clear that the functions characterized by Proposition 1.6.9 depend on the ultrafilter Q, hence the difficulty in determining their existence. Numerosity presents some other interesting challenges and open questions, but we will remand them to Section 3.4 of this paper, when we will have at our disposal more mathematical tools. 1.7 Extensions of sets The Ω-limit is a mathematical instrument that allows the construction of hyperreal numbers out of real numbers. In the same spirit, a notion of Ω-limit can also be given for functions ϕ : Pfin(Ω) → P(R): as a result, we will be able to “extend” the subsets of R into subsets of R ∗ by using the Ω-limit. We start by defining what we intend as the Ω-limit of a sequence of sets. Definition 1.7.1. Let {Eλ}λ∈Pfin(Ω) be a family of nonempty subsets of R. We define lim Eλ = λ↑Ω lim ϕ(λ) : ϕ(λ) ∈ Eλ λ↑Ω and lim c∅(λ) = ∅ λ↑Ω In the same spirit, if E is a subset of Ω, we define lim Eλ = lim ϕ(λ ∩ E) : ϕ(λ ∩ E) ∈ Eλ λ↑E λ↑Ω Thanks to Transfer Principle, the definition of limλ↑Ω Eλ can be given by an “almost everywhere” formulation. More precisely, we will show that Proposition 1.7.2. If {Eλ}λ∈Pfin(Ω) is a family of nonempty subsets of R, then 18 lim Eλ = lim ϕ(λ) : ϕ(λ) ∈ Eλ a.e. λ↑Ω λ↑Ω
1.7. EXTENSIONS OF SETS Proof. By hypothesis, for all λ ∈ Pfin(Ω) we can pick eλ ∈ Eλ. Now, suppose that ϕ(λ) ∈ Eλ a.e.: then, if we define a function ϕ ′ ϕ(λ) if ϕ(λ) ∈ Eλ (λ) = eλ if ϕ(λ) ∈ Eλ we have that ϕ ′ (λ) = ϕ(λ) almost everywhere and, by Transfer of equality, we conclude limλ↑Ω ϕ ′ (λ) = limλ↑Ω ϕ(λ). Now observe that, by construction, limλ↑Ω ϕ ′ (λ) ∈ limλ↑Ω Eλ, so the thesis follows. Thanks to Definition 1.7.1, we can now define two kinds of extensions of sets: the first, more generic, is the natural extension. Definition 1.7.3. The natural extension of a set E ⊆ R is given by E ∗ = lim cE(λ) = lim ϕ(λ) : ϕ(λ) ∈ E λ↑Ω λ↑Ω where cE(λ) is the function identically equal to E. Then, we can define another extension, more restrictive than the one we’ve already seen, but with more interesting properties: the hyperfinite extension. Definition 1.7.4. The hyperfinite extension of a set E ⊆ R is given by E ◦ = lim λ↑E λ It’s worthy to explicitate this definition: E ◦ = lim ϕ(λ ∩ E) : ϕ(λ ∩ E) ∈ λ ∩ E λ↑Ω From the two definitions above, it’s clear that E ⊆ E ◦ ⊆ E ∗ . Now, we want to study some elementar properties of those extensions of sets, and their behaviour under some basic set operations. The first property we want to show is that equality is preserved under natural and hyperfinite extension. Proposition 1.7.5. Let E, F ⊆ R. E = F if and only if E ∗ = F ∗ if and only if E ◦ = F ◦ . Proof. If E = F , then it’s trivial that E ∗ = F ∗ and E ◦ = F ◦ . Now, we want to show that, if E = F , then E ∗ = F ∗ and E ◦ = F ◦ . Let’s suppose that E ⊆ F , so that we can choose x satisfying x ∈ E and x ∈ F . In particular, we have that, if ϕ(λ) ∈ F for all λ ∈ Pfin(Ω), then ϕ(λ) = x and so x ∈ F ◦ and x ∈ F ∗ , so that E ◦ ⊆ F ◦ and E ∗ ⊆ F ∗ . This allows to conclude E ◦ = F ◦ , as desired. In the next two Propositions, we will see that the only sets for which it holds the equality chain E = E ◦ = E ∗ are finite sets. We will split the proof of this statement in two parts: at first, we will prove that E = E ∗ if and only if E is finite, and then that E ◦ = E ∗ if and only if E is finite. 19
Page 1: UNIVERSITÀ DEGLI STUDI DI PAVIA FA
Page 5 and 6: Contents 1 Ω-Calculus 1 1.1 Infin
Page 7 and 8: Introduction Borknagar La mia vita
Page 9 and 10: INTRODUCTION notions of set theory
Page 11 and 12: Chapter 1 Ω-Calculus Mathematics
Page 13 and 14: 1.1. INFINITESIMAL AND INFINITE NUM
Page 15 and 16: 1.2. SUPERREAL FIELDS Thanks to inf
Page 17 and 18: 1.3. THE AXIOMS OF Ω-CALCULUS In
Page 19 and 20: 1.4 First properties of Ω-limit 1
Page 21 and 22: 1.4. FIRST PROPERTIES OF Ω-LIMIT
Page 23 and 24: 1.4. FIRST PROPERTIES OF Ω-LIMIT
Page 25 and 26: 1.6. NUMEROSITY limλ↑Ω |λ|
Page 27: 1.6. NUMEROSITY In the previous par
Page 31 and 32: 3. E ∗ ∪ F ∗ = (E ∪ F ) ∗
Page 33 and 34: Example 1.7.13. Let E = R: by defin
Page 35 and 36: 1.8. EXTENSIONS OF FUNCTIONS result
Page 37 and 38: 1.9. TOWARDS A MODEL FOR Ω-CALCUL
Page 39 and 40: 1.9. TOWARDS A MODEL FOR Ω-CALCUL
Page 41 and 42: 1.10. FILTERS AND ULTRAFILTERS Proo
Page 43 and 44: 1.10. FILTERS AND ULTRAFILTERS clea
Page 45 and 46: is a commutative ring and F is an u
Page 47 and 48: 1.11. A MODEL FOR Ω-CALCULUS We c
Page 49 and 50: 1.12. REASONING IN THE MODEL Remark
Page 51 and 52: 1.12. REASONING IN THE MODEL Proof.
Page 53 and 54: Chapter 2 Selected topics of Ω-Ca
Page 55 and 56: Example 2.1.4. Let’s consider the
Page 57 and 58: 2.1. CONVERGENCE AND CONTINUITY Def
Page 59 and 60: 2.2. TOPOLOGY BY Ω-CALCULUS Now,
Page 61 and 62: 2.2. TOPOLOGY BY Ω-CALCULUS z ∈
Page 63 and 64: 2.2. TOPOLOGY BY Ω-CALCULUS Proof
Page 65 and 66: 3.1. HYPERFINITE SUMS relations wit
Page 67 and 68: the Ω-integral by showing that [
Page 69 and 70: 3.3. Ω-INTEGRAL AND RIEMANN INTEG
Page 71 and 72: [a,b) ◦ 3.4. NUMEROSITY AND LEBES
Page 73 and 74: 3.4. NUMEROSITY AND LEBESGUE MEASUR
Page 75 and 76: Chapter 4 Ω-Theory A mathematicia
Page 77 and 78: 4.2. THE AXIOMS OF Ω-THEORY Ω-A
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4.3. GENERALIZING CONCEPTS Proposit
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4.4. TRANSFER PRINCIPLE BY Ω-THEO
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4.4. TRANSFER PRINCIPLE BY Ω-THEO
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4.5. A MODEL FOR Ω-THEORY b = lim
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4.5. A MODEL FOR Ω-THEORY In our
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Chapter 5 The formal version of Tra
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5.1. LOGIC FORMALISM FOR SET THEORY
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5.1. LOGIC FORMALISM FOR SET THEORY
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• (σ ∨ σ ′ ) ≡ ¬(¬σ
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Chapter 6 An application: Non-Archi
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6.2. NON-ARCHIMEDEAN PROBABILITY Re
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6.3. FIRST CONSEQUENCES OF NAP AXIO
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6.4. FAIR LOTTERIES BY NAP Remark 6
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6.4. FAIR LOTTERIES BY NAP Using Nu
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6.5. INFINITE SEQUENCE OF COIN TOSS
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Chapter 7 Further ideas We have man
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7.2. DIMENSION BY Ω-THEORY We’v
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7.3. NUMEROSITY AND LEBESGUE MEASUR
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and, by Transfer of inequalities, w
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7.7. THE DIRAC DISTRIBUTION In orde
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we obtain that 7.8. THE NATURAL EXT
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7.8. THE NATURAL EXTENSION OF NUMER
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BIBLIOGRAPHY [16] Imre Lakatos, Pro
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RINGRAZIAMENTI Ricordo con riconosc
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