Is there a homomorphism of finitely presented groups $f:G\to H$ and an element $h\in H$ such that the statement "$f^{-1}(h)$ is empty" is independent of ZFC?



Empty preimage under homomorphism of finitely presented groups independent of ZFC
The answer is yes, as a consequence of [my answer to your other question](https://mathoverflow.net/a/390821/1946).


Namely, in that answer, we have a finite group presentation $H$ and a word $h$ such that the question $h=1$ in $H$ is independent of ZFC. So if we take $G$ to be trivial and $f:G\to H$ the unique homomorphism, we have the statement "$f^{-1}(h)$ is nonempty" being independent of ZFC.


The general lesson of that answer supplies also an answer to Benjamin Steinberg's comment here concerning a confusion between computable undecidability and ZFC or logical undecidability. The general lesson, which I argue on the other post, is that every computably undecidable enumerable decision problem is saturated with logical undecidability. So the two notions of undecidability are actually intimately connected.


