If F is a (potentially infinite) family of finite nonempty sets, why doesn't it have a choice function even in ZF? By definition, finite sets have a bijection with an initial set of natural numbers, so can't you use the bijection for each set in F to select the element that maps to 1? And for that matter, if F is a family of countable sets then each C in F has an injection to the naturals, which are well-ordered. So can't you use the injection to map each C in F to its member that maps to the minimal element of the range of the injection from C to the naturals?
2026-04-08 19:38:53.1775677133
Why doesn't family of finite nonempty sets have choice function?
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