Re: Axiomatizations of set theory

On Mon, 08 May 2006 05:09:15 GMT, Nam Nguyen <namducnguyen@xxxxxxx> said:
JB wrote:
[...] both Z and ZF have both axioms and an axiom schema.

to which CM [mildly] protested:
Well, just to be clear ;-) ZF (and Z) don't "have" axiom schemas;

with an explanation:
they are just theories, sets of sentences closed under consequence.

Of course CM's protest is technically correct; theories don't "have"
have axiom-schemas, though they might have *axiom-schema axioms*
(which is what JB apparently meant but mis-stated slightly). I just
felt CM's explanation didn't really fix the mis-statement.

True enough, we often say, e.g., "ZF has the following axioms" to
indicate a conventional axiomatization of ZF. My niggling complaint was
mostly directed at the idea that schemas are themselves part of ZF.

.

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