It is a known theorem that for a model of $ZF$, $M$, if $M\models AC$ and $G$ is a $P$-generic filter over $M$, for some $P\in M$, then $M[G]\models AC$.
On the other hand, it is long known that other axioms, for example $GCH, CH, V=L, ...$ are not preserved by such extensions.
In a paper of Monro [1] the first paragraph speaks on a question by Dana Scott about why do some axioms get preserved by generic extensions and other not; it also says that there is an [almost obvious] equivalence between preservation in generic extensions, and preservation in Boolean-valued models.
The paper goes on to prove that some restricted versions of the axiom of choice are not preserved by generic extensions (and in a recent phone call it was explained to me how to break weak choice principles such $DC_\kappa$ quite easily in relatively simple generic extensions of models of $ZF$).
Lastly the paper says nothing about an answer to the question above, nor it cites any resource for a possible answer. However, the paper is quite old now, and in the past three decades some progress might have occurred.
Is there an answer for Scott's question: which axioms are preserved by all generic extensions?
Edit: If there is no "simple" and uniform answer to the above question, is there a possible answer to why is the Axiom of Choice preserved in generic extensions, while restricted versions are not?
Bibliography:
- G. P. Monro, On Generic Extensions Without the Axiom of Choice. The Journal of Symbolic Logic Vol. 48, No. 1 (Mar., 1983), pp. 39-52
