FOM: Re: categorical cardinals (in two senses)

[Larry Stout]

Two of the early papers in topos theory (Tierney's "Sheaf Theory and the
Continuum Hypothesis" and Marta Bunge's work on topoi and Souslin's
hypothesis, both from the early 1970's) showed ways that topos theoretic
methods could capture results related to questions about cardinals.

I think that categoricity in the model theoretic sense is not the kind of
property which appeals to the topos theorist's foundational sense:  if you
are looking for the wide variety of different settings in which mathematics
can be done, then when you find a theory which has only one model which is
unique up to isomorphism you conclude that you have been too specific.
Category theorists (at least this one) are not looking for "The" foundation
for mathematics, but they are intersted in "foundations of mathematics".

There may well be problems developing a theory of large cardinals in a topos
theoretic setting because most topoi do not satisfy the axiom of choice.
Because so much of recent category theory has focused on issues related to
computation-- in some sense dealing with small things-- there may be little
interest in developing large cardinals in a topos setting.

Perhaps the upcoming panel on the need for new axioms for mathematics will
convince me that the issue needs to be developed.  I tend to be interested
in the variety which comes from assuming less rather than the specificity
which comes from assuming more.

Lawrence Neff Stout