Re: Question: What conditions are sufficient to prove a subset does not exist?

On Wed, 25 Jul 2007 22:10:53 -0000, Scott <ToaTerra@xxxxxxxxx> wrote:


Hi: I had a question I hope the community can help with. Let S be any
set. Let A = { x in S : x notin f(x) } where f:S -> P(S). What is
necessary and sufficient to show A does not exist? (Please ignore for
the moment that the axiom schema of separation says A does exist ...

We *can't* ignore it, since the existence of A -at least in ZFC minus
substitution- depends on the axiom schema of separation. Actually, the
very symbol you use to denote A /shows/ this. In general:

{x e z : phi(x)}

is the set y such that x e y <-> x e z & phi(x). (The axiom of
extensionality ensures that there actually is only ONE such set.)


as I'm trying to determine what would be the conditions necessary
to prove it does not exist.)

Huh? Well, of course, the axiom schema of substitution implies the
axiom schema of separation. But I guess that's not what you meant?


F.

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