Schanuel conjecture and its set theoretic status 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? 
 A: You need to be a bit careful how you state it. Remember that Con(ZF) is a $\Pi_1^0$ statement, and that is independent of ZF.
I think what you're looking for is Shoenfield absoluteness. This implies that $\Pi_2^1$ statements are independent of certain axioms that can be added to ZF. Amongst these axioms are the axiom of choice and the generalised continuum hypothesis. Hence if you have a proof of Schanuel's conjecture that assumes choice and CH, then there must be a proof using only the axioms of ZF.
It's also the case that if a $\Pi_2^1$ statement holds, then it also holds in any transitive models of ZF.
A: 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.
