TITLE: Schanuel conjecture and its set theoretic status
QUESTION [5 upvotes]: I have once been told that Schanuel's conjecture (which is a transcendence statement about tuples of complex numbers and their exponentials) is set theoretically absolute, which means that its truth (or falsity) is independent of the set theoretical model (or assumptions), because it is a $\Pi^1_2$ statement. Is this correct?

REPLY [7 votes]: This is an extremely sloppy reference to the Shoenfield absoluteness theorem.
This theorem says that given an arithmetical statement about the natural numbers which is not "too complicated" (first-order statements are not too complicated, but not just them), then the statement is true in every two transitive models of $\sf ZFC$ which have the same ordinals.
This doesn't mean that the truth value is independent of $\sf ZFC$. It just means that we cannot prove independence using the two "usual" tools of set theory:

Forcing, which does not change the ordinals of a model, nor destroy its transitivity; and
Inner models, which are by definition a subclass of the model which is transitive and contains all the ordinals.

There might be other ways to prove independence, e.g. compactness arguments of some sort. But things like compactness usually destroy nice properties of models of $\sf ZFC$, and the absoluteness theorem no longer applies. On the other hand, we are less interested in not-nice models of $\sf ZFC$ in general, which is why set theorists focus on the two methods above, rather than other methods.
So the conclusion is that if we can force the truth value of the Schanuel conjecture, then we actually proved it.