The Axiom of Choice and its Discontents

Transcription

1 The Axiom of Choice and its Discontents Jacob Alexander Gross University Of Pittsburgh February 22, 2016 Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

2 Outline 1 What is the axiom of choice? 2 Origins of the Axiom 3 Why it is Obviously True 4 Hilbert and Brouwer Duke it Out 5 Current Status 6 The Godel-Gentzen Translation 7 The Banach-Tarski Paradox 8 Godel and Cohen Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

3 What is the axiom of choice? An axiom is a proposition taken to be true without proof one must begin somewhere. Axioms are supposed to be self-evident The axiom of choice is often considered the most controversial axiom in the history of mathematics. Specifically it says: Let C be a nonempty collection of sets. Then, there exists a function f (called a choice function) defined on C such that f (S) S, for all S C. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

4 Equivalents The Well-Ordering Theorem says that any set can be well-ordered. Zorn s Lemma says that if (A, <) is a poset such that every chain in A is bounded above in A, then a A there exists an maximal element greater than a. Both of these are equivalent to the axiom of choice. The Hausdorff Maximality Principle (HMP) and the Tukey-Teichmuller lemma are also equivalent. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

5 Equivalents The Well-Ordering Theorem says that any set can be well-ordered. Zorn s Lemma says that if (A, <) is a poset such that every chain in A is bounded above in A, then a A there exists an maximal element greater than a. Both of these are equivalent to the axiom of choice. The Hausdorff Maximality Principle (HMP) and the Tukey-Teichmuller lemma are also equivalent. Joke: The axiom of choice is obviously true, the well-ordering theorem is obviously false and nobody understands Zorn s lemma. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

6 Origins of the Axiom The axiom of choice was first formulated in 1904 by Ernst Zermelo (in a different form using what he called coverings ). The Well-Ordering Theorem was fundamental in Cantor s transfinite arithmetic and was an open problem at the time (proved by Zermelo s invokation of the AC). The French pre-intuitionists Lebesgue, Baire and Borel rejected this on the belief that a mathematical object exists just in case it can be explicitly defined. There were also worries that the introducing the axiom may cause set theory to be inconsistent. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

7 Why it is Obviously True In 1904 Russell shows the that AC implies an empty product may be non-empty. Russell contends AC is a multiplicative axiom, to ensure the product of nonzero cardinals is nonzero. The axiom of choice is required to prove that every vector space has a basis. The axiom of choice is required to prove that every proper ideal is contained in a maximal ideal. There are many other crucial mathematical results relying on the axiom of choice (Tychnoff s theorem, all isotropic spaces are contained in Lagrangian spaces, etc.). Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

8 Hilbert and Brouwer Duke it Out David Hilbert was a renowned mathematician heavily involved in the foundations of mathematics. Early in Hilbert s career he proved Hilbert s basis theorem which proved the existence of a basis for finite invariants. Hilbert did not prove this by explicitly saying what the basis was, only that it would be contradictory for it not to exist. Paul Gordan (the King of Invariant Theory ) allegedly said Hilbert s paper is not mathematics, it is theology. Hilbert saw this kind of non-constructive argument as absolutely central to mathematics. Hilbert endorsed the axiom of choice as an essential mathematical principle. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

9 Hilbert and Brouwer Duke it Out Brouwer was a tremendous mathematician aside from his involvement in foundations (Brouwer s fixed point theorem, the degree of a map, etc.). His paper On the Unreliability of Logical Principles openly challenged the law of excluded middle. Without it, proving an existential means giving an explicit construction. Hilbert was vehemently opposed saying Taking the law of the excluded middle from the mathematician is like prohibiting a boxer the use of his fists. It is provable that without the law of the excluded middle, we do not have choice either (there is a caveat here involving something called a locally cartesian closed category if you are interested in this talk to me afterwards). Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

10 Current Status of the Axiom of Choice Weyl, Markov, Bishop and Martin-Lof followed in Brouwer s footsteps. Today most mathematicians accept the axiom of choice and the law of the excluded middle. Your Professor will let you use them on tests and homework... you might even get extra credit if you use the axiom of choice in a clever way. Due to the controversial nature of AC, if a theorem depends on choice it is customary to point this out. Zorn s Lemma probably the most popular form of AC for use in pure mathematics. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

11 The Godel-Gentzen Translation The Godel-Gentzen translation gives a way to translate classical proofs into constructive ones. It can be made precise by recursion on first-order formulas but we will not do that here. Basic idea of GG translation Classical Mathematician I proved Hilbert s basis theorem! Constructive Mathematician No you did not! You proved Hilbert s basis theorem cannot possibly be false! Classical Mathematician Whatever. To me, its the same thing. Constructive Mathematician Well to me its not but I understand what you re saying. For this reason intuistionistic logic is more expressive than classical logic. The intuitionistic existential and disjunction are more specific than their classical counterparts. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

12 The Banach-Tarski Paradox 1 The Banach-Tarski paradox was first stated in The original statement was that one could take a ball, slice it up into only 6 pieces and then, using only rigid motions, put these pieces back together into two balls of equal size and volume as the first. 3 More generally it is a theorem that one can cut up a ball the size of a pea and put the pieces (rigidly) back together to form a ball the size of the sun. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

13 The Banach-Tarski Paradox 1 The Banach-Tarski paradox was first stated in The original statement was that one could take a ball, slice it up into only 6 pieces and then, using only rigid motions, put these pieces back together into two balls of equal size and volume as the first. 3 More generally it is a theorem that one can cut up a ball the size of a pea and put the pieces (rigidly) back together to form a ball the size of the sun. 4 This result relies on the axiom of choice. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

14 The Banach-Tarski Paradox 1 The Banach-Tarski paradox was first stated in The original statement was that one could take a ball, slice it up into only 6 pieces and then, using only rigid motions, put these pieces back together into two balls of equal size and volume as the first. 3 More generally it is a theorem that one can cut up a ball the size of a pea and put the pieces (rigidly) back together to form a ball the size of the sun. 4 This result relies on the axiom of choice. 5 We either reject the axiom of choice or say the ball was sliced up into a pile that cannot be assigned a measure in any meaningful sense. 6 Indeed there are models of analysis where all sets are measurable... so long as they exclude choice. Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

15 Godel and Cohen In 1940 Kurt Godel put to rest the suspicions that introducing the axiom of choice caused set theory to be inconsistent. He proved that the axiom of choice is consistent with von Neumann-Godel-Bernays set theory (a more conservative version of ZF set theory). acob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

16 Godel and Cohen In 1940 Kurt Godel put to rest the suspicions that introducing the axiom of choice caused set theory to be inconsistent. He proved that the axiom of choice is consistent with von Neumann-Godel-Bernays set theory (a more conservative version of ZF set theory). In 1963, Paul Cohen proved that AC is also consistent with set theory. acob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

17 Godel and Cohen In 1940 Kurt Godel put to rest the suspicions that introducing the axiom of choice caused set theory to be inconsistent. He proved that the axiom of choice is consistent with von Neumann-Godel-Bernays set theory (a more conservative version of ZF set theory). In 1963, Paul Cohen proved that AC is also consistent with set theory. The axiom of choice is completely independent from the other axioms.. acob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13

18 References I [1] Banach, S. and Tarski, A. Sur la decomposition des ensembles de points en parties respectivement congruentes. Fund. Math. 6, , [2] Bell, John L., The Axiom of Choice, The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), URL = < [3] Machover, Mosh (1996) Set Theory, Logic and Their Limitations. Cambridge. [4] MacLane, S. and Moerdijk, I. Sheaves in Geometry and Logic A first introduction to topos theory. Springer Verlag, [5] Strombers, K. The Banach-Tarski Paradox. Amer. Math. Monthly. 86 3, Jacob Alexander Gross (University of Pittsburgh)The Axiom of Choice and its Discontents February 22, / 13