UPDATE: I edited the answer by adding details and adding a reference. In particular, I specialized from an arbitrary field to the complex numbers in response to John's comment.

Here's an example from commutative algebra. The projective dimension of a module M is defined as the minimal length of a projective resolution of M. Let CC mean the complex numbers, and let S be the ring CC[x,y,z] ℂ[x,y,z] and M be the module CC(x,y,z). ℂ(x,y,z). Then the projective dimension of M is undecidable in ZFC. More specifically, the projective dimension of M is 2 if the continuum hypothesis holds, and it is 3 if the continuum hypothesis fails.

This follows from Barbara Osofsky's work (MR0548131). MR0548131); see Theorem 2.51 of Homological Dimensions of Modules. She seems to have a huge number of results which would be relevant to this question.

UPDATE: I edited the answer by adding details and adding a reference. In particular, I specialized from an arbitrary field to the complex numbers in response to John's comment.

Here's an example from commutative algebra. The projective dimension of a module M is defined as the minimal length of a projective resolution of M. Let CC mean the complex numbers, and let S be the ring k[x,y,zCC[x,y,z] and M be the module k(x,y,z)CC(x,y,z). Then the projective dimension of M is undecidable in ZFC.

I learned this in conversation with my advisor onceMore specifically, so I don't the projective dimension of M is 2 if the continuum hypothesis holds, and it is 3 if the continuum hypothesis fails.

This follows from Barbara Osofsky's work (MR0548131). She seems to have a reference to point huge number of results which would be relevant to this question.But I'd love it if someone suggested one!