FOM: truth and provability

[Martin Davis]

At 06:27 PM 11/8/00 +0100, Kanovei wrote:

> > 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

It's not I who go around in circles.


> > 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.
>

My point which I would have thought was exquisitely clear is that this form 
gives an absolutely precise sense in which, for anyone who accepts the 
soundness of T (even with respect to sentences of a very simple logical 
form), a sentence can be seen to be true although not provable in T.

Of course for someone who imagines that using the word "myth" will demolish 
ordinary mathematical practice, none of this counts.

I will not participate any longer on this thread.

Martin


                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
                          (Add 1 and get 0)
                        http://www.eipye.com