Since the Compactness theorem of FOL is equivalent to the ultrafilter Lemma, which implies Hahn-Banach, the implication is clear to me. I was more just wondering if there is a nice direct proof? I saw in another thread that Compactness very nicely proves the existence and uniqueness of Algebraic closures, so I thought maybe there's a neat proof of Hahn-Banach as well. I just can't think of a good way to axiomatize the statement, as the class of topological spaces and normed spaces is not axiomatizable, and I'm guessing the class of locally convex topological vector-spaces is neither using similar arguments. Can anyone help out? Thanks in advance!
2026-03-25 22:04:07.1774476247
Proof of Hahn-Banach theorem from Compactness theorem of FOL
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