The class of binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$ forms a Borel set, neither closed nor open.

• Can you show it's not $F_\sigma$ or $G_\delta$?

• Is it actually complete for level $\omega$ of the Borel hierarchy?

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?

The class of binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$ forms a Borel set, neither closed nor open. Is it in some sense complete for finite levels of the Borel hierarchy?

The class of binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$ forms a Borel set, neither closed nor open.

• Can you show it's not $F_\sigma$ or $G_\delta$?

• Is it actually complete for level $\omega$ of the Borel hierarchy?