# Empty preimage under homomorphism of finitely presented groups independent of ZFC

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?

• It seems to me that any homomorphism between finitely presented groups is computable, whatever it means. Do I miss something? – YCor Apr 21 at 17:48
• We can take $f$ a morphism taking everything to identity. Then for appropriate $H$ we have that ZFC can't tell whether $H$ is nontrivial, so it can't check the nontriviality on all generators. – Wojowu Apr 21 at 17:50
• @Wojowu isn't the preimage of the identity always non-empty? – Oniqa Apr 21 at 17:51
• @Oniqa But ZFC won't be able to prove that these elements are or aren't identity. – Wojowu Apr 21 at 17:53
• @Wojowu but the identity is certainly getting sent to the identity – Oniqa Apr 21 at 17:54

## 1 Answer

The answer is yes, as a consequence of my answer to your other question.

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.

• But the independence from ZFC thing seems to make some assumptions about cardinals – Benjamin Steinberg Apr 22 at 2:11
• @BenjaminSteinberg The $\Sigma_1$-correctness was only for the very general fact. Meanwhile, the example at the end of my other answer produces a ZFC independent instance assuming only Con(ZFC), which is required in order to have anything independent of ZFC. – Joel David Hamkins Apr 22 at 8:35