Does set forcing preserve the failure of AC?

Question

Let $V$ be a model of ZF + $\neg$AC. Can there be a forcing extension of $V$ that satisfies AC?

Answer 1

Sometimes, often, perhaps, the answer is yes. But sometimes it is not.

Andreas Blass defined "Small Violations of Choice" which turned out to be exactly equivalent to "There is a set forcing which forces AC". This principle holds true in any and all symmetric extensions, given by a set forcing, and indeed it turned out to be equivalent. Namely, if $V$ satisfies SVC, then there is an inner model $W$ which is a model of $\sf ZFC$, and $V$ is a symmetric extension of $W$ by a set forcing.

Finally, it is also possible for SVC to fail. In the Gitik's model, Morris' model, and the Bristol model SVC fails. Indeed, the first two satisfy that you cannot extend the model, at all, by forcing or otherwise, to a model of $\sf ZFC$ without adding new ordinals.