FOM: categorical cardinals (in two senses)

[Stephen G Simpson]

Larry Stout writes:

 > 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.

The papers that you cite are simply translations into the topos
setting of earlier set-theoretic work of Cohen and
Jech/Tennenbaum/Solovay, proving the consistency and independence of
CH and SH from ZFC, via forcing.  (Compare also my previous posting,
where I mention the well known connection between forcing and
intuitionism.)

In any case, Palmgren and I were talking not so much about cardinals,
but rather about *large* cardinals, i.e., cardinals whose existence
cannot be proved in ZFC.  My specific question concerned weakly
compact cardinals.  Is there anything in the topos setting analogous
to weakly compact cardinals?  Is it elegant?  Is it fruitful?

-- Steve