Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
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Well, the proof itself goes through some other equivalents as well. So it might be slightly easier to understand the full circle:
$(1)\implies(2)$ is a well-known argument of transfinite induction, and $(2)\implies(3)$ is the classical use of Zorn's lemma. $(3)\implies(4)$ can be intuitively comes from the fact that linear combinations are finite, so we can define a vector space and from its basis choose the finite subsets of the family. Finally, $(4)\implies(1)$ can be handwavingly described as reiterating the subset process until we are left with just singletons from which the choice is canonical.
The proofs themselves of course require a lot more details, and the above intuitive and handwaving explanation is far from sufficient to fully get them. But that's the main intuition, I think.