Re: Osius' set theory

[Wilkins E B <elwood@essex.ac.uk>]

Paul Taylor wrote:

>
> Gerhard Osius's work was significant in 1974 as one of the ways in which
> it was shown that toposes do the same thing as set theory.

Osius showed that well-pointed topoi do essentially the same thing as set theory.
I have yet to find any adequate analogy between what elementary topoi in general
do and set theory. The problem being, as far as I can see, that classical set
theory finds it very difficult to describe the rather more subtle behaviour
occurring in the initial topos. Whence the standard set theoretic account of
topos logic includes the assumption that bound variables denote, which make the
semantics of elementary topoi almost identical to those of well-pointed ones (for
instance every proper subtype of 1 is treated as though it is empty!).

>   In fact Osius's
> is, so far as I am aware, the only work that discusses *set* theory -
> Benabou, Joyal, Mitchell, Lawvere etc showed that toposes do the same thing
> as some form of *type* theory.  Of course, it is the latter that
> mathematicians actually use when they claim to be using set theory
> as foundations ...

In practice this is true, but not if they're interested in actually studying the
foundations of maths. My (set minded) colleagues point out (quite forcibly) that
unless something is expressible in ZF (say) then they find it hard to accept, a
view that appears to be widespread in the theoretical end of computer science.
Now I have the practical problem of convincing people that what I do
constructively is better than what they do with their "bound variables denote"
assumptions, and an axiomatic set theory equivalent to the initial topos would
make this much easier: they could go away contentedly believing that I'm really
doing set theory (while I can be happy with the knowledge that they're doing
typed theory).

There are potential applications to category theory from producing a set theory.
For instance one might speculate that this set theory could replace the rather
cumbersome Freyd cover in the proof of the constructive properties of the initial
topos. A host of other speculations can be formulated ...

Elwood


--
Dr Elwood Wilkins                               e-mail: elwood@essex.ac.uk
Senior Research Officer                         tel:    (+44) (0)1206 872336
Department of Computer Science                  fax: (+44) (0)1206 872788
University of Essex, Colchester, Essex, UK