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- 1. On the Axiom of Choice Flora Dellini Marco Natale Francesco Urso
- 2. Preliminary ›For every set S, a set U is a subset of S if, for every item in U, this item belongs to S too. ›For every set S, we define the set of the parts of S, P(S), as the set of all the possible subsets of S. Example: 𝑆 = 1,2,3 𝑃 𝑆 = {∅, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , {1,2,3}}
- 3. The Axiom of Choice (AC) › The Axiom of Choice is a statement about the existence of a certain kind of functions. › A choice function is a function which selects an item from a subset of a given set. › AC claims that, for every group of subsets of S, there exists a function of choice which selects a particular item from every given subset. ∀𝑆, ∀𝑈 ⊆ 𝑃 𝑆 ∅, ∃𝑓: 𝑈 → 𝑆 such that ∀𝑋 ∈ 𝑈, 𝑓(𝑋) ∈ 𝑋
- 4. AC: Examples with finite sets When a set is finite, everything is trivial. The existence of f is not disputable: we can actually show and build it! 𝑆 = 1,2,3,4,5 𝑈 = { 1 , 1,2,3 , 2,3 , 3,4,5 } ⊆ 𝑃(𝑆) 𝑓: 𝑈 ⊆ 𝑃(𝑆){∅} → 𝑆 𝑓 1 = 1 ∈ 1 𝑓 1,2,3 = 3 ∈ 1,2,3 𝑓 2,3 = 2 ∈ 2,3 𝑓 3,4,5 = 5 ∈ 3,4,5
- 5. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks › Let’s suppose there is a shop with infinite pair of shoes, we want to built an infinite set containing a shoe for each pair. › A Turing machine (i.e. a personal computer) can choose between right or left shoe because it can distinguish them. For example we can build the set of all right shoes. › Can we do the same with infinite pair of socks?
- 6. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks › A machine can not do it while a man could do. › The reason is: right and left shoes can be distinguished due to this feature. A man can do this, a machine can do it too. › It is not possible to choose right or left socks, because there are no such things as right or left socks! › As a consequence, in principle, a compuer can not build the wanted set.
- 7. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks ›Could a man build a set from a single sock for each pair? ›Actually, it is impossible, because we would die before we can name all the items in the set. ›But is it possible in principle? How can we distinguish between the socks of a pair? ›We do this by choosing “this one”, without any criterion but our free will.
- 8. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks ›As human beings, it is in our everyday experience that we can distinguish between two socks by calling «this one» or «that one». ›Is it admissible that in maths it is also possible to act in such a way? ›The answer is far from trivial!
- 9. The Way of Formalism › The mathematical concept of Set raised up in XIX century by Georg Cantor (1845-1918). › Cantor’s idea of Set was a “collection” of “objects” which satisfy certain properties (e.g. “the set of all odd numbers”, “the set of all the right shoes”). › Paradoxes arise from a “too free” use of the concept of “property”.
- 10. The Way of Formalism › Russell Paradoxes: the set of all the sets which don’t contain themselves. › The problem is in the semantic. › D. Hilbert’s “Formalism” school proposed the reduction of mathematics to a pure “formal game”. In this way, Mathematics would have been stripped of all its “human components”.
- 11. The Way of Formalism › Zermelo and Fraenkel, following Hilbert’s intuition, began to develop the Formal Set Theory (also known as ZF). › In according to the formalistic concept of mathematics, the semantic in ZF is “eliminated” reducing the concept of proprieties to pure syntactic formulas, computable in principle by a machine. › The nature of Sets is so implicitly defined by syntactic formulas.
- 12. The Way of Formalism ›The validity of a formula must be determined through “propositional calculus”, an absolutely formal procedure which can be, theoretically, implemented on an ideal machine. ›E.g. instead of saying “there is the empty set”, we shall write the following formula: ∃𝑦 ∀𝑥 (𝑥 ∉ 𝑦)
- 13. The Axioms of Zermelo-Fraenkel 1. Axioms describing implicitly the concept of Set (Regularity and Extensionality Axioms). 2. Axiom of Existence. The unique axiom of existence in ZF is the Axiom of Infinity, which asserts that there exists an infinite set. 3. Axioms of Individuation, which allow us to “individuate” (i.e. build) new sets starting from ones already known (Axiom of Power Set, Axiom of Union and Axiom of Replacement).
- 14. Is AC compatible with ZF? › Being compatible means that, if we add AC to ZF, we cannot deduce a theorem and its negation. › ZF claims to describe all mathematical universe: “…with regard to ZF it’s hard to conceive of any other model”. P. Cohen. › Because we would like to proceed in maths as we do with socks, that is by choosing items as we want to, the compatibility of AC with ZF is highly desirable.
- 15. ZF does not disprove AC “Inside only ZF, it’s not possible to prove that AC is false” (Gödel, 1938). Main steps of proof: 1. Gödel added another axiom to ZF (“every set is constructible”), obtaining the stronger theory ZFL. 2. Gödel proved that ZFL is consistent. In this stronger theory he proved that every set can be well-ordered, that is a demonstration of AC. So, ZFL → AC.
- 16. ZF does not disprove AC 3. If, by contradiction, AC is false in ZF, it has to be false also in ZFL. But we have just seen that AC is true in ZFL! So AC could not be disproved in ZF. ZF does not disprove AC! This does NOT mean that AC is true in ZF !
- 17. ZF does not imply AC In 1960 Cohen has completed Gödel’s demonstration about independence of AC from ZF. So ZF does not imply AC.
- 18. Independence of AC from ZF As a consequence, we can add or remove AC from ZF as we like. So, its presence is actually a preference of the mathematician who can want a “richer” or “poorer” theory.
- 19. Theorems we lose without AC • Every non empty Vectorial Space has a base – i.e. imagine that, in the classic Euclidean 3D space, you don’t have the 𝑖, 𝑗, and 𝑘 vectors you use to build every other vector. • Every field has an algebraic closure – i.e. imagine that you could not define the complex numbers
- 20. What does it mean? “Simply” that there are sets whose volume is not invariant under translation and rotation. Strange! Isn’t it? This is why some mathematicians worry about the Axiom of Choice Decomposition of a ball into four pieces which, properly rotated and traslated, yield two balls Counterintuitive effects of AC: Banach-Tarski Paradox
- 21. Counterintuitive effects of AC: Zermelo’s Lemma This statement is equivalent to the Axiom of Choice: Every set S can be well ordered As a consequence, we could «well order» ℝ - i.e. defining an order in ℝ such that every subset of ℝ has a minimum. This order relation is strongly counterintuitive, as it implies that sets like (0,1), without a minimum – 0 ∉ (0,1) do not exist.