TITLE: Axiom of Choice in Logic
QUESTION [5 upvotes]: The completeness and compactness theorems of first-order logic are well known to be equivalent to the ultrafilter lemma. Are there any theorems of logic that are similarly equivalent to the full axiom of choice? A slightly lesser question: does the ultrafilter lemma suffice for intuitionist logic, which (as I understand it) needs an infinite set of truth values?

REPLY [3 votes]: Another equivalent:

Given a set $S$ of sentences of first order logic, and a subset $B$ of $S$ such that $B$ is consistent, there is a consistent subset $A$ of $S$ such that $A$ is maximal among the consistent subsets of $S$, and such that $B\subseteq A.$

Result due to Klimovsky (iirc). See Rubin & Rubin Theorem 8.4.