Re: Implementable Set Theory and Consistency of ZFC

Virgil wrote about:

No set exists unless (1-4) require it to exist.

In article <6f1f$473022ce$82a1e228$24739@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

No set exists unless (1-4) require it to exist.

Yes. That's what I'm trying to say all the time.

THEN YOU MUST MAKE IT AN AXIOM.

I don't know how to say it precisely. What I do know
is that such an axiom would no longer be considered
a first-order axiom. It would be second-order.

Sorry. I don't know what it all means.

Roughly speaking, it is 2nd order because it quantifies over sets, not merely over elements.

Huh? I thought that, in ZFC, _all_ elements were _sets_!

Han de Bruijn

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