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:

David C. Ullrich wrote:

Saying he's wrong does not prove that he is. Your "proof" that 5-8
follow from 1-4 is simply not a correct proof - that's a fact,
independent of anything anyone may have said here.

Yes, you've declared that repeadedly, by _dogma_. Are you the Pope of
Mathematics or what? Or is it true that I'm not allowed to enter the
Sacred Rites of you and your companions?

No, you can prove him wrong. Using only the axioms (1)-(4) and any
standard first order logic, give a derivation of (5)-(8). That's how
you prove what you've claimed.
This is *not* what you've done. Instead, you've given a particular
model of (1)-(4) and shown that (5)-(8) are true in that model. But
that does not show that (5)-(8) are theorems of (1)-(4).

No. There is not _one_ model in the first place. There are at least four
of them: integer array, bitmap, character string, natural. Computational
Set Theory is about sets that can be computed, represented in a machine,
it's not "just a model" of a far more superior Theory of Sets. Bringing
set theory back to earth is what I want. And I've succeeded a great deal
in doing so. (It has been a surprise to me that nine out of the ten ZFC
axioms still stand and have not collapsed with demanding computability.)

Even assuming that there are four models rather than one, so what?
The point is that you have *not* given a proof of (5)-(8) using only
axioms (1)-(4). Thus, you have not done what you said.

Even if you show that there are seven models for (1)-(4) in which
(5)-(8) are also true, you haven't done what you said. Even if you
show there are *infinitely* many models satisfying this condition, you
haven't done what you said.

So, perhaps you should either do what you said or change your claim.

No. Because _nothing_ sensible ever counts as a proof in your conception
of mathematics. Let those who can not keep up do not halt the parade ..

Han de Bruijn

.

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