seeing more math on the TL recently. i haven't mathposted in a long time but i'm still happy to field math questions from twitter peeps, just reply here or tag me or w/e. big meaty philosophical questions like "wtf are real numbers anyway" would be particularly fun for me rn
Conversation
Why is the axiom of choice not obvious - why does it require an axiom?
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it’s not obvious at all! it implies the existence of all sorts of crazy stuff like a well-ordering of the reals that is provably impossible to write down constructively (since it’s consistent with ZF - set theory minus choice - that there isn’t one)
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it only sounds obvious if you phrase it as something like “the product of nonempty sets is nonempty” because you’re intuitively considering finite collections of nonempty sets. once you get to, say, uncountable collections of very complicated sets it’s a lot less so