Harvey Friedman <friedman-hyEunQjjh3V1yXs8afm8B5Vzexx5G7lz@xxxxxxxxxxxxxxxx>
Tue, 24 Oct 2006 22:00:59 -0400
Wrote:
> FOL (through its practically driven variants) is the only
> appropriate vehicle for the foundations of mathematics. I
> doubt if many people would disagree with this. The only
> issue is how to go about carefully formulating just what
> this means.
I want to say I entirely agree with this. Harvey and I have notably
disagreed over what kind of axioms in FOL ought to be taken as
foundational. But after thinking about it a whole lot for years, and
seriously considering various options, I entirely agree that FOL is the
vehicle. For one example, precisely in my field of topos theory, the
higher-order internal logic is itself a manifestation of more
fundmental properties of each topos expressible in FOL.
I will not try to give my reasons -- because I also agree with Harvey
that there remains a serious task of fully formulating the claim.
best, Colin
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