Re: Implementable Set Theory and Consistency of ZFC

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

Jesse F. Hughes wrote:

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

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?

There is, perhaps, a way in which your model is canonical. It seems
to me that it is a minimal model and, as you say, (5) is true in that
model. It simply does not follow that (1)-(4) entail (5).

And, as we have said, if (1)-(4) *did* entail (5), then so would
(1)-(4) + (X).

--
"Destiny is a funny thing. Once I thought I was destined to become
Emperor of Greenland, sole monarch over its 52,000 inhabitants. Then
I thought I was destined to build a Polynesian longship in my garage.
I was wrong then, but I've got it now." -- The Tick
.

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