[FOM] Categoricity in f.o.m.

[Harvey Friedman]

On Mon, Apr 4, 2016 at 2:00 PM, Arnon Avron <aa at tau.ac.il> wrote:
> So why not calling a theory "weakly categorical" (say) if
> it has a single minimal model that can be embedded in any other
> model, and view weakly categorical theories as being
> (mainly) about their unique minimal model?
>
> Do  we really need the usual notion of categoricity?
> I think that weak categoricity is good enough.
>
> Needless to say, PA, and various predicative set theories
> are weakly categorical.
>
> Arnon Avon
Y
our notion is an important and relevant notion also. It should be used
as well, in any discussion of the coherence and robustness of the
usual f.o.m. It is my impression that coherence and robustness are not
exactly the hallmarks of some of the seriously alternative foundations
being discussed.

My postings on categoricity in f.o.m. were inspired by some posting of
John Baez in a different forum, where he didn't see why logicians
emphasized categoricity of set theory since "we already know from
Goedel that categoricity is impossible". I pointed out that Baez is
talking about first order categoricity only. Hence my postings, where
I do NOT want to bring in second order logic wholesale, but just the
idea of respecting (sub)sets and the like.

Harvey Friedman