[FOM] What are foundations for?

[Robert Solovay]

Martin,

Harvey Friedman showed that Borel Determinacy can't be proved in
Zermelo set theory. Martin's proof used the existence of aleph_one
iterations of the power set operation.

-- Bob Solovay

On Fri, Feb 24, 2012 at 12:45 PM, Martin Davis <martin at eipye.com> wrote:
> From my perspective, the argument posed as set theory vs category
> theory is taking place in a pre-Goedel world when foundations were
> supposed to be the single solid rock supporting the corpus of
> mathematics.
>
> In the post -Goedel world, that's a remote and forgotten Eden. As
> Goedel himself pointed out already in his 1933 lecture, instead of a
> single system we are confronted by a hierarchy of systems, and as we
> rise in the hierarchy, more and more previously unproved propositions
> become provable. These include Pi-0-1 propositions that each may be
> interpreted as stating that some polynomial equation has no integer
> roots. In this world what is important is systems that can be used to
> calibrate the strength of propositions.
>
> The proof of FLT famously not only used categorical methods, but
> freely used Grothendiek universes. This immediately raised the
> questions: is it provable in PA, in second-order arithmetic, or what?
> Category theory is a wonderful tool, but what use is it here?
>
> Goedel, in effect, conjectured in his Gibbs lecture that the Riemann
> Hypothesis is not provable in PA.  Can category theory hope to resolve
> this?
>
> I believe (someone correct me if I've got this one wrong) Tony Martin
> proved that Borel determinancy is provable from the Zermelo axioms but
> not from n-th order arithmetic for any n. What use would categorical
> methods be in making that kind of fine discrimination?
>
> Martin
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