For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it is consistent to have $BS(V)=\emptyset$ or $\vert BS(V)\vert>1$.

Now let $\mathfrak{BS}$ be the class $\{BS(V): V\text{ a vector space}\}$. My question is:

What restrictions on $\mathfrak{BS}$ can be proved in ZF?

(The question I *want* to ask is "What are the possible values for $\mathfrak{BS}$?", but since without choice there is no canonical picture of what the cardinalities 'are' I'm not even sure how to phrase that.)

*Of course, the question can be asked in general for generating sets of arbitrary algebraic structures. One more question I would love to know the answer to, but which I think is too broad, is How different is the situation when we ask the same question in this much greater generality?*

I am asking this question for vector spaces over all fields, but I would also be interested in answers for vector spaces over specific (classes of) fields, especially (classes of) well-orderable fields.

To help make this question less hopelessly broad, let me ask a specific sub-question, the answer to which I'm sure is "no" (but I can't prove it):

Is it consistent with ZF that $\mathfrak{BS}$ is closed under finite intersections?

In general, what are the "best" (i.e., most structurally rich - closed under lots of operations) kinds of algebraic structure $\mathfrak{BS}$ can have?