Intuition for "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?

Question

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?

Answer 1

Caveat lector: There are plenty of easily understandable statements in mathematics, whose proofs are far from trivial or intuitive. This is not exactly the case here, but it is not the case that it's not the case here either. I will try to give a handwavy explanation as to why this works, but this is far from trivial.

Well, the proof itself goes through some other equivalents as well. So it might be slightly easier to understand the full circle:

1. The axiom of choice.
2. Zorn's lemma.
3. Every vector space has a basis.
4. The axiom of multiple choice (every family of non-empty sets of size at least $2$ has a function which chooses from each set a finite proper subset).

$(1)\implies(2)$ 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.