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wuncidunci comments on Godel's Completeness and Incompleteness Theorems - Less Wrong
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The reason why compactness is not provable from ZF is that you need choice for some kinds of infinite sets. You don't need choice for countable sets (if you have a way of mapping them into the integers that is). You can get a proof of compactness for any countable set of axioms by proving completeness for any countable set of axioms, which can be done by construction of a model as in Johnstone's Notes on Logic and Set Theory p. 25.