Re: Countable models of ZFC

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*.
Everything that's *decided* is decided the SAME way in ALL
models of the axioms from which it was decided. EVERYThing
else IS WHOLLY dependent on "models" if it is to have *a*
(as opposed to "either") truth-value AT ALL. If a first-order
sentence
has a model independent truth-value then you DON'T say that it is
"true" (even if it is) or "false" (even if it is): you say that IT IS
A
*THEOREM* or that it is PROVED, which is of course MUCH
STRONGER than being "true".


The truth of some sentences of the language of ZFC are relative to
a model.

But that by definition occurs only in the case where those selfsame
sentences are, in some OTHER models, FALSE.

Therefore you simply CAN'T say that they are "true".
I am AGREEING with Herbzet here.


The relevant point here is that there is ONE axiom-set where
all this can be worked around, and that is first-order PA.
We think we understand N *prior* to this axiomatization and
we think we know that N *is ONE* model of this axiomatization.
Because this model is standard, we can ABBREVIATE
"true in the standard model of PA" with JUST PLAIN "true".
We SHOULDN'T have, but we could and we did.
Because of this colossal pedagogical error, people
like Rupert now think that something ANALOGOUS *must*
be available with OTHER RICHER first-order axiom-sets, like
ZFC. Since they can't pin down an individual standard model
(or maybe they could, but if they could, then it would be, in
contradistinction to PA's standard model, which is minimal,
MAXIMAL), they are now trying to resort to a class of standard
models. But they are not doing a coherent job of it, at least not
locally.


.

Relevant Pages

  • =?iso-8859-1?q?Re:_Torkel_Franz=E9n_is_dead?=
    ... an assertion about a particular natural number, ... PA, as a first-order language, or even as a first-order theory, ... ALL interpretations, ... standard model, DESPITE the fact that there is nothing going ...
    (sci.logic)
  • Re: Sorry Godel - All Truths are Provable
    ... Rupert wrote: ... I was talking about truth in the standard model for arithmetic. ... Which contains predicates P, which are essentially sets. ...
    (sci.logic)
  • Re: Provable in T?
    ... Rupert wrote: ... Is it the case that G_T is true in the standard model? ... Do we need to say "If T is consistent then G_T is true in the standard ... if T is sound it will be. ...
    (sci.logic)