Is there a finitely generated computably presentable group $G$ on generator set $A$ and a computable function $f$ from first-order formulas to words on $A$ such that $\mathsf{ZFC}\vdash\sigma\leftrightarrow\tau$ iff $f(\sigma)$ and $f(\tau)$ represent the same element in $G$?



Is there a finitely generated group with the same structure as ZFC?
The relation $\text{ZFC}\vdash\varphi\leftrightarrow \psi$ is a $\Sigma\_1^0$-definable equivalence relation on the set $\mathcal L$ of formulas in the language of set theory. It is a corollary of Theorem 3.2 of Neis-Sorbi's ["Calibrating word problems of groups via the complexity of equivalence relations"](https://arxiv.org/pdf/1609.03371.pdf) that there is a finitely generated computably presentable group with generator set $A$ whose word problem, viewed as an equivalence relation $\sim$ on the set $W$ of words on $A$, is $\Sigma\_1^0$-universal. As a consequence there is a reduction from the former equivalence relation to the latter, and this just means that there is a computable function $f : \mathcal{L}\to W$ such that $f(\varphi) \sim f(\psi)$ if and only if $\text{ZFC}\vdash\varphi\leftrightarrow\psi$, which is what you want.


