Re: Implementable Set Theory and Consistency of ZFC

On Oct 31, 7:57 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Jesse Hughes wrote:

Whatever. Again, I ask: Do you agree that "There is at most one empty
set" is a theorem of ZFC, *even though* it uses only one of the axioms
of ZFC?

Yes.

The proof of existence uses an instance of separation and uniqueness
uses extensionality. Or, if you have an empty set axiom, then, yes,
just one axiom.

But aside from those pedantic matters, now you see our point, right?
The proof of a theorem might use only certain premises but it is still
a theorem from any larger set of premises, just as this is an example:
the proof of the existence of a unique empty set needs to use only
certain axioms, not all of them, but the existence of a unique empty
set is also a theorem of the entire set of axioms.

MoeBlee


.

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