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
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