# FOM: truth and provability

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Wed Nov 8 12:27:15 EST 2000

```> Date: Tue, 07 Nov 2000 14:38:27 -0800
> From: Martin Davis <martin at eipye.com>

> Can I be the only fom-er getting tired of this discussion going round and=
> round in circles

That you go in circles is simply because you do not like
to accept that there is no any mathematical statement that
can me qualified as TRUE by reasons different from its being
PROVED mathematically, or, saying the same differently,
that the existence of such a sentence can be demonstrated
only assuming "ontological" existence of some kind of
"standard model" of PA.

> following variant form of Goedel's theorem:
>
> For any consistent axiomatic extension T of Robinson's Q (hence certainly
> PA and ZFC are included) there is a formula with one free variable G(x),
> where numerals representing natural numbers may be substituted for x, such
> that:
> 1. for each numeral n, G(n) is provable in T.
> 2. (Ax)G(x) is not provable in T.

This is just another form of the same.
What you suggest amounts to the following.

THEOREM (ZFC or PA).
There is a formula G(x) such that, if Con ZFC, then
(1) ZFC-Prov G(n) for any n in N
(2) not-ZFC-Prov (An)G(n).
PROOF
See Goedel.
END OF PROOF

REMARK
G(x) can be taken to say: "x does NOT code a proof of 0=1 in ZFC".
In this case, ZFC-Prov G(n) implies G(n) (assuming Con ZFC).
Therefore, (1) can be replaced by
(1') G(n) for any n, hence, (Ax)G(x) "is true".
END OF REMARK,

Which leads us exactly to the same true-non-provable myth as in
the earlier course of discussion.

V.Kanovei

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