If we suppose the Axiom of Choice or Zorn's Lemma, are any infinite dimensional vector spaces isomorphic as vector spaces?

Question

My question is, if we suppose the Axiom of Choice or Zorn's Lemma, are any infinite dimensional vector spaces isomorphic as vector spaces? If not, are all countable vector spaces isomorphic as vector spaces? Or are all uncountable (or, say, $\aleph_1$ cardinality) vector spaces isomorphic as vector spaces?

Edit: I guess I've sort of answered part of my own question here, because an isomorphism must be a bijection, and there is no bijection between countable and uncountable sets. But I'm still curious about the other parts of the question . . .

Answer 1

Assuming the axiom of choice, two vector spaces over a fixed field are isomorphic if and only if they have bases of the same cardinality.

So if $V$ and $W$ are two vector spaces over some field $\Bbb F$, then they are isomorphic if and only if they they bases of the same cardinality.

Now, even with or without the axiom of choice, not all countable vector space are isomorphic. Note that $\Bbb Q^n$ and $\Bbb Q^m$ are isomorphic if and only if $m=n$, this is because finite dimensional vector spaces have the "usual theory" without appealing to choice. So you might want to be more careful in that formulation.

Answer 2

There are infinitely many cardinal numbers for a base, all $\aleph_\alpha$, $\alpha \in \operatorname{Ord}$. If we have two vector spaces $V$ and $W$ over some field $\mathbb{K}$, Zorn tells us they both have a base. They are isomorphic iff these bases have the same cardinality. This is clearly necessary and sufficient.