Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

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).

I don't agree. Infinity "adds" something to the system that makes it far
less transparent. Things that were (almost) trivial for finite sets, are
NO longer trivial for infinite sets. That's exactly the reason why some
theorems HAVE to appear AS axioms. As I've said many times, these axioms
are indeed theorems for the finitary sets in ZFC, but not for infinitary
sets, and I'm not the one who is making this up. Choice is a well known
example.

Han de Bruijn

.

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