On 2023-12-16 18:26:39 +0000, Ross Finlayson said:

If you count the axiom schema of separation as just one instead of a schema, ZF definitely is finitely axiomatized. (Also the other schemas have finite versions, i.e. their statements about filling the entire space of the higher-order.)

Goedel's results are about finitely axiomatized theories.

Gödel's incompleteness theorem (and related results) only require
that there is a finite procedure that can determine whether a
sentence is an axiom or not. It is not necessary that they can
be expressed with a finite number of schemas.

Mikko

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)