The collection of


*binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$*


forms a Borel set, neither closed nor open -- assuming Con(ZFC).


* Can you show it's not $F\_\sigma$ or $G\_\delta$?
* Is it actually complete for level $\omega$ of the Borel hierarchy?



Models of ZFC and the Borel hierarchy
@FrançoisG.Dorais commented:



> 
> By reading the axioms (i.e. every axiom is satisfied, as a formalized statement) it's a $\Pi^0\_{\omega+1}$ set (since satisfying a first-order sentence is $\Pi^0\_n$ for some $n$). I think that it's not $\Pi^0\_n$ for $n<\omega$ amounts to the fact that ZFC is not finitely axiomatizable.
> 
> 
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