Re: Implementable Set Theory and Consistency of ZFC

In article <b2040$47274d5b$82a1e228$5003@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

David C. Ullrich wrote:

On Mon, 29 Oct 2007 15:35:33 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

The fact that Infinity X is an axiom of standard ZFC makes it necessary,
it seems, to include the axioms (5-9), in order to make infinite sets
make more "look alike" finite sets.

Wow. I mean really, wow. You claim that 5-8 follow from 1-4, but
somehow if we add 9 then we also need to include 5-8 as axioms?

Yes. Infinitary set theory needs more axioms than finitary set
theory.

If you can prove (5) from axioms (1)-(4), then you can prove it from
axioms (1)-(4) + (X). Adding a new axiom does not invalidate existing
proofs.

With (1)-(4), a universe of finite sets is created. Within that universe
I can prove that (5) is valid. But then comes (X) and there is no way to
create any set compatible with (X) in my universe. Hence (5) can not be
proved for those sets you call infinite (I call them: not existing). Is
that so difficult to comprehend?

What you claim would seem to require an additional axiom that says that
anything not explicitely required is forbidden.
.

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