—-

Notes:

Several versions of "the" Banach-Tarski theorem, and

- what axioms they use

— James Taylor

- whether they're important to other branches of maths

- definitions are like axioms, but

— 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.