Peter Humphries

2008 ASC report:



Several versions of “the” Banach-Tarski theorem, and - what axioms they use — possible sets of axioms — e.g. non-measurability & Hahn-Banach Theorem — James Taylor - whether they’re important to other branches of maths - definitions are like axioms, but — When mathematicians unify a theory, they discard definitions, and that has a lot of consequences, even though they tend to not say anything about those consequences. — Problems which get blamed on AC could alternatively be blamed on — definitions (e.g. Lesbegue measure), or — ZF (see above e.g. James Taylor) - example: Importance of topological invariant definition in proving the Poincar’ Conjecture - What I’m saying is also often said by mathematicians BUT not formally

Axiom of Choice could be a case study for this general point:

When e.g. a topologist wants to prove some theorem in topology, they’ll be able to prove it in any reasonable set theory.

BUT in some set theories they’ll need auxiliary axioms that they might not like (e.g. AC) and in other set theories they might not.

  • A really simple example is that in some set theories infinity is an axiom and in others it’s a theorem.