Re: Countable models of ZFC

On Oct 6, 12:15 am, george <gree...@xxxxxxxxxx> wrote:
On Oct 4, 1:51 am, herbzet <herb...@xxxxxxxxx> wrote:

Rupert wrote:
If you're going to take the view that you can never understand any
sentence unless you've specified what model it's relativized to,
you're going to tie yourself in knots.

Rupert himself is already tied in knots around this.

In the first-order paradigm, there is basically NO SUCH THING
as "truth". ALL truth of ALL first-order sentences is model-
theoretic.
If you want model-independent truth then you have to GET it
via something that IS NOT truth, NAMELY, *provability*.

I don't agree with this view. And I think the view that there is only
model-theoretic truth is incoherent, because then you have to ask what
the semantics of model theory is, and you are left without a
foundation.

But if you don't like my views about the semantics of mathematics,
then fine, just interpret every mathematical statement I make
formalistically. There is still no cause to start ranting and raving
and shouting and insulting when I make perfectly ordinary mathematical
statements.

We can have a discussion about the semantics of mathematics if you
like. You're saying you can't give a semantics for the first-order
language of set theory which isn't model-theoretic. I don't agree.
This is because philosophically I object to the idea of the set-
theoretic universe being there "all at once". You might like to read
about the "Putnam semantics" for the first-order language of set
theory, in which the first-order language of set theory is interpreted
in a language with modal operators. It's discussed in Geoffrey
Hellman's "Mathematics Without Numbers."


.

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