Are there any purely semantic proofs of the compactness theorem that don't use the full axiom of choice?
139337
Ultrafilter Lemma implies Compactness/Completeness of FOL
28629
Is the axiom of choice necessary to prove that closed points in the Zariski topology are maximal ideals?
49220
Open Sets of $\mathbb{R}^1$ and axiom of choice
65308
Axiom of Choice: Family of non-empty sets mutually disjoint or not?
3708
How can I prove that compact subset iff sequentially compact in second countable topological spaces?
115675
Strange consequences of Axiom of Choice in Zermelo set theory
33563
Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)
19209
Why is compactness in logic called compactness?
84378
Does axiom of foundation/regularity protect against Russell-like paradoxes?
121847
Don't we need the axiom of choice to choose from a non-empty set?
9158
Proving that a propositional theory of any cardinality has an independent set of axioms
127575
Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice
88099
Not $\sigma$-compact set without axiom of choice
40682
What's going on with "compact implies sequentially compact"?
84562
Proof using only field axioms
157490
Statements equivalent to the Axiom of Choice
62936
Proving equivalencies of two versions of the Axiom of Choice