Re: Implementable Set Theory and Consistency of ZFC

On Thu, 01 Nov 2007 13:52:35 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

On Thu, 01 Nov 2007 10:03:34 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

MoeBlee wrote:

On Oct 31, 1:59 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

Jesse F. Hughes wrote:

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).

How do you "know" that? Has the Pope told you, by dogma, that it is so?

No, you royal ignoramus, it's PROVEN as a basic property of the
deductive system. It's the property of monotonicity of deduction. And
we PROVE it.

Yeah, yeah. Then why are (5-9) required in ZFC, once Infinity has become
an axiom of it? Why is e.g. Choice provable in (ZFC-Infinity) and not in
common ZFC?

This has been explained several times. If by ZFC-Infinity you mean
ZFC without the axiom of infinity then Choice is _not_ provable
in that syatem. "ZFC without the axiom of infinity" is not the
same as "ZFC plus the negation of the axiom of infinity" - choice
_is_ provable in ZFC plus the negation of the axiom of infinity.

Sure. (ZFC - Foo) is not the same as (ZFC - Foo + ~Foo). Quite clear.
Tip, hint: how can Foo be "denied" iff Foo is just plain nonsense in
the first place? But there is more.

"ZFC without Infinity" does _not_ deny Infinity. It does not
assume Infinity. There's a big difference.

Because (apples without pears) is not the same as (apples without pears
and (no pears)). And I am so stupid that I don't understand this? Wow!

That's not a correct analogy.

"1-4" is not the same as "1-4 are true, and 5-8 are false".

But ZFC does not consist of "ZFC plus the negation of Infinity"
plus more axioms (it _is_ equal to "ZFC without Infinity" plus
more axioms).

Han de Bruijn


************************

David C. Ullrich
.

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