The axiom of choice is already true in ZF for finite families of sets without the full axiom of choice: https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf
Confusingly (to me, at least; I haven’t spent the time to understand this), AoC and Banach-Tarski are also true generally in the constructible universe, which is nice as a model of ZF, but reasonably defined sets are still measurable, so you don’t get the Banach-Tarski paradox if you only use them: https://math.stackexchange.com/questions/142499/are-sets-constructed-using-only-zf-measurable-using-zfc
Also note that Banach-Tarski uses a different kind of (and larger) infinity, specifically the continuum (real numbers) to model objects. If the universe is discrete but infinite in extent, then you can still avoid Banach-Tarski.
Thanks! I have replaced “They allow for the Banach–Tarski paradox” by “Some types of infinity allow for the Banach–Tarski paradox”.
The axiom of choice is already true in ZF for finite families of sets without the full axiom of choice: https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf
Confusingly (to me, at least; I haven’t spent the time to understand this), AoC and Banach-Tarski are also true generally in the constructible universe, which is nice as a model of ZF, but reasonably defined sets are still measurable, so you don’t get the Banach-Tarski paradox if you only use them: https://math.stackexchange.com/questions/142499/are-sets-constructed-using-only-zf-measurable-using-zfc
Also note that Banach-Tarski uses a different kind of (and larger) infinity, specifically the continuum (real numbers) to model objects. If the universe is discrete but infinite in extent, then you can still avoid Banach-Tarski.
Thanks! I have replaced “They allow for the Banach–Tarski paradox” by “Some types of infinity allow for the Banach–Tarski paradox”.