Re: what makes it true?

Torkel Franzen <torkel@xxxxxxxxxx> writes:

> Esa A E Peuha <esa.peuha@xxxxxxxxxxx> writes:
>
> > But then GC is not just true but also provable.
>
> What is provable in PA is "if GC is undecidable in PA then GC is
> true".

Yes. It is also provable in PA that "if GC is true then GC is provable
in PA" (by Gödel's completeness theorem). Therefore it is provable in
PA that "if GC is undecidable in PA then GC is provable (and hence
decidable) in PA", and finally from this it is provable in PA that "GC
is decidable in PA".

> If it is provable in some theory T that GC is undecidable in
> PA, then GC is provable in T.

True, but irrelevant to what I wrote.

> PA itself does not prove any statement
> undecidable in PA.

I never claimed it did.

--
Esa Peuha
student of mathematics at the University of Helsinki
http://www.helsinki.fi/~peuha/
.

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