Re: Implementable Set Theory and Consistency of ZFC

On Oct 23, 12:02 pm, MoeBlee <jazzm...@xxxxxxxxxxx> 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.

I'm curious as to whether some of 5-9 are derivable from some of
the other axioms plus the _negation_ of Infinity (~X).

In particular, it is often pointed out that the Axiom of Choice is not
needed for finite sets. I believe it was MoeBlee (though it might have
been someone else) who pointed out that it's redundant to write
ZFC-Infinity, but just write ZF-Infinity instead. In other words, 9 is
derivable from 1-8 + ~X.

I also attempted to prove Powerset (7) from the others, much
earlier in this thread. I don't know how accurate the proof was, but
I certainly couldn't avoid Replacement (6). So it's possible that
7 can be derived from 1-6 + ~X.

Of course, this is still far from HdB's claim to derive all of 5-9
from
1-4 + ~X.

.

Loading