FOM: topos theory

[Michael Thayer]

Vladimir Kanovei writes:

>
>
>However mathematiciands working with *bridges*
>(which by Simpson means real analysis in its
>classical form) pretend to work with THE reals,
>i.e. something uniquely determined by the very
>basic nature of mathematics (if not by the
>physical reality).
>
>It is a great advantage of set theory that it
>grounds and supports this approach.
>
>Does the topos theory do this ?
>
>But the expected answer amounts to the following:
>*take a "category of sets" or a "topos well pumped
>by ZFC" and go ahead*.
>If so then the total picture is not inspiring.
>
I am not sure what the objection to this situation is, although I gather it
is similar to Steve's objection (which I also don't get).  Is the problem
that one needs to do "extra stuff" to get going when starting with a topos??
Let me give an example:  There is a set theory, KF (described among other
places in Forster's "Set Theory with a Universal Set") which can either be
extended to Z or extended to NF.  Would you say that KF does NOT "ground and
support" analysis because of the underlying ambiguity in the nature of "set"
in KF?  (After all, NF and Z(FC) have somewhat different notions of set,
don't they?)
I am trying to see exactly what is the nature of the problem Vladimir and
Steve have with Colin's statement that you can do ZFC kinds of things in the
appropriate topos.  I don't think either of them are DISPUTING  Colin's
claim, but I don't see why they view it as unresponsive or uninspiring
either.  Any help would be greatly appreciated.
Michael Thayer