Independence of AC

Set Theory · Axiom Academy

The Axiom of Choice cannot be proved or disproved from ZF 1. Gödel's Consistency Result (1940) Kurt Gödel proved that if ZF is consistent, then ZF + AC is also consistent. He did this by constructing the "constructible universe" L. Every set in L is "constructible" from simpler sets by a definable process L also satisfies AC (and even the Generalized Continuum Hypothesis!) If ZF has a model, then ZF + AC has a model (namely L) This shows that AC cannot lead to a contradiction unless ZF itself is already inconsistent. Adding AC is "safe." 2. Cohen's Independence Result (1963) Paul Cohen proved the converse: if ZF is consistent, then ZF + ¬AC is also consistent. He invented the revolutionary technique of "forcing." Use forcing to carefully add new sets to M, creating an extension M[G] M[G] still satisfies ZF but can be made to violate AC For example: add an infinite family of sets with no choice function Cohen's work was revolutionary - forcing became a fundamental tool in set theory. He received the Fields Medal in 1966 for this achievement. Combining Gödel's and Cohen's results, we conclude that AC is independent of ZF: ZF ⊬ AC (ZF doesn't prove AC, by Cohen) ZF ⊬ ¬AC (ZF doesn't prove ¬AC, by Gödel) Therefore: AC is independent of ZF The independence of AC is analogous to the independence of Euclid's parallel postulate in geometry. Just as we can have Euclidean or non-Euclidean geometries, we can have mathematics with or without AC.

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