It is well-known that if you force over a model of $\mathsf{ZFC}$ you get another model of $\mathsf{ZFC}$. Also basic is that the Axiom of Choice itself plays no real part in the mechanics of forcing (so that one can force over models of $\mathsf{ZF}$ to get another model of $\mathsf{ZF}$, though probably not one in which the Axiom of Choice holds). I am curious whether any of the more common weak choice principles are preserved by forcing. That is, for which $\mathsf{X}$ is it the case that $M[G] \vDash \mathsf{ZF}+\mathsf{X}$ whenever $M \vDash \mathsf{ZF}+\mathsf{X}$, $\mathbb{P} \in M$, and $G$ is $\mathbb{P}$-generic over $M$?
In particular, I would be interested in the following:
- The Axiom of Countable Choice, $\mathsf{CC}$: for any sequence $( X_n )_{n \in \mathbb{N}}$ of nonempty sets, the product $\prod_{n \in \mathbb{N}} X_n$ is nonempty.
- The Axiom of Dependent Choice, $\mathsf{DC}$: if $R$ is a binary relation on a nonempty set $X$ such that for each $x \in X$ there is a $y \in X$ such that $x \mathrel{R} y$, then there is a sequence $( x_n )_{n \in \mathbb{N}}$ in $X$ such that $x_n \mathrel{R} x_{n+1}$ for each $n \in \mathbb{N}$.
- The Ultrafilter Theorem, $\mathsf{UF}$: given any filter $\mathcal{F}$ of subsets of a nonempty set $X$, there is an ultrafilter $\mathcal{U}$ on $X$ with $\mathcal{U} \supseteq \mathcal{F}$.
I'm pretty sure that first is not preserved, I'm pretty agnostic on the second, and I'm inclined to think that the last may be preserved. This is more of a gut feeling than anything else so it can be safely ignored.
None of these things need to be preserved in general.
There Monro proves that all three can be violated by forcing. In fact, we can show that if $\sf DC$ or even $\sf AC_\omega$ are preserved by generic extension (and it's enough to consider well-orderable forcings), then $\sf AC_{WO}$ holds. You can find some details in this MathOverflow thread.
We can show that $\sf DC$ is preserved under proper forcing (see my paper with David Asperó, Dependent Choice, Properness, and Generic Absoluteness), and in general it is also not hard to show that $\sf DC_{<\kappa}$ is preserved under $\kappa$-closed forcings (well, also $\kappa$-proper forcings, I suppose, but those are not well-behaved as proper forcings are with respect to $\omega$). But we don't have a nice condition for $\sf AC_\kappa$ (without $\sf DC_\kappa$, that is) or the Boolean Prime Ideal theorem (or even weaker versions such as the Ordering Principle).
On a more positive side, we can define a family of principles called Kinna–Wagner Principles, where $\sf KWP_\alpha$ states that every set can be injected into $\mathcal P^\alpha(\eta)$ for some ordinal $\eta$. Then we can show that if $\alpha$ is a limit (or $0$) then $\sf KWP_\alpha$ is preserved in forcing extensions, and in general forcing can violate at most one level in the Kinna–Wagner hierarchy. You can find more about this in one of my recent papers, Guide to the Bristol model: Gazing into the Abyss.
Finally, since we mentioned MathOverflow, this and this are both relevant to your question.