Is this:
for every vector space $V$, if $B$ and $C$ are bases of $V$, then there is a bijection: $B\to C$ iff the axiom of choice holds
true? Or, perhaps, if axiom of choice is replaced by something weaker?
2026-04-04 15:19:03.1775315943
Axiom of choice and vector space bases
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It is true that the axiom of choice is true if and only if every vector space has a basis to begin with. However the statement that if there is a basis then every two bases have the same cardinality is much weaker than the axiom of choice.
My question on MathOverflow received a very nice answer with references, that in fact the Boolean Prime Ideal theorem (which is strictly weaker than the axiom of choice) implies this, and probably even less.