I was reading [this answer](https://mathoverflow.net/a/32914/146831), which says that:



> 
> In his Master's Thesis, Merlin Carl has computed a polynomial that is solvable in the integers iff ZFC is inconsistent. A joint paper with his advisor Boris Moroz on this subject can be found at <http://www.math.uni-bonn.de/people/carl/preprint.pdf>.
> 
> 
> 


Note that the link is dead. Emil Jeřábek provided an alternate link here: [*A polynomial encoding provability in pure mathematics (outline of an explicit construction)*](https://doi.org/10.36045/bbms/1366306724).


That phrase "solvable in the integers iff $\mathsf{ZFC}$ is inconsistent". If $\mathsf{ZFC}$ is inconsistent, then of course the polynomial is solvable in the integers - every statement in the model is true! So it seems to be just a fancy way of saying that the polynomial is not solvable in the integers.


I believe I'm missing some subtleties here, so I would like to have someone address this confusion of mine.



What does it really mean for a polynomial to be solvable in $\mathbb{Z}$ iff $\mathsf{ZFC}$ is inconsistent?
Edit: After I wrote this, the paper was posted in the commentary. What I wrote below is what is meant, however they use $\mathsf{GBC}$ instead of $\mathsf{ZFC}$. Since these two theories are equiconsistent and have the same first order consequences, this technical difference is immaterial.


However, when people say things like this there are a number of ways to interpret it, none of which are trivial. If I had to guess, probably what is meant is something like ``there is a polynomial $p(x)$ (often concretely computed) and $\mathsf{ZFC}$ proves ($\exists x \in \mathbb Z$ $p(x) = 0$ iff $\mathsf{ZFC}$ is inconsistent) ". By the second incompleteness theorem even if $\mathsf{ZFC}$ is consistent it cannot prove this fact so the equivalence is not trivial. Moreover, there are models on $\mathsf{ZFC}$ which think $\mathsf{ZFC}$ is consistent, and in such models there are no integer solutions to $p(x)$ and other models of $\mathsf{ZFC}$ which think that $\mathsf{ZFC}$ is inconsistent and in such models some (necessarily nonstandard) integer will satisfy $p(x)$.


The point is usually that the kinds of coding that lead to the incompleteness theorems can in fact be coded into surprisingly concrete polynomials. In the case of $\mathsf{PA}$ there are many examples of such coming from MRDP/Matiyasevich's theorem. In the wikipedia article on diophantine equations this is mentioned: <https://en.wikipedia.org/wiki/Diophantine_set>, see ``further applications".


