Re: Implementable Set Theory and Consistency of ZFC

On Oct 24, 12:50 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx>
wrote:
MoeBlee wrote:
On Oct 23, 4:23 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

1. Extensionality 5. Specification X. Infinity
2. Empty set 6. Substitution
3. Pairing 7. Power Set
4. Union 8. Foundation
9. Choice

And, as I've said, in this "model", only (1-4) are necessary as axioms,
because (5-9) appear as theorems. And (X) is not part of the "model".

I just saw Virgil's post, which made me realize I overlooked that you
said 5-9 are theorems of 1-4!

You're wrong. None of 5-9 is derivable from all of 1-4.

What you might mean is that in a certain model of 1-4, we have that
5-9 are true in that model. But that does not ENTAIL that any of 5-9
are theorems of 1-4.

You are wrong. Read the article.

I'm wrong on what point? The article doesn't prove that all of 1-4
entail any of 5-9. Again, that 5-9 are true in a particular model
along with 1-4 is not a proof that 1-4 entail any of 5-9. Do you
understand that or not? Are you familiar with (or at least know of)
the independence proofs of these axioms?

MoeBlee

.

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