Re: Implementable Set Theory and Consistency of ZFC

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

Jesse F. Hughes wrote:

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.

Choice is (I think) a theorem in the theory of finite sets. The
theory of finite sets includes an axiom explicitly stating that every
set is finite. This is not the same as your theory, which includes no
such axiom.

I will say it once more. I am typing this slowly, since I don't want
you to miss anything I say. If you have a proof in a theory
consisting of axioms (1)-(4), it is also a proof in the theory
consisting of axioms (1)-(4)+(X).


--
Jesse F. Hughes
"Mathematicians who read proofs of my results seem to basically lose
some part of themselves, like it rips at their souls, and they are no
longer quite right in the head." -- James S. Harris, Geek Cthulhu
.

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