FOM: topos theory qua f.o.m.; a quote from Mac Lane

[Colin McLarty]

Steve Simpson writes:

>As I said in my posting of yesterday:
>
> > My question is:
> > 
> >   What additional axioms are appropriate to add to topos theory to get
> >   enough real analysis to build bridges?  (By "enough real analysis to
> >   build bridges" I mean the standard undergraduate material on
> >   ordinary and partial differential equations, power series, Fourier
> >   analysis, etc., complete with applications, examples, etc.)
>
>Note that ZFC is not mentioned in this question.

        That post was a reply to my post of Thu, 15 Jan 1998 in which I said:

>>>you can do real analysis in any topos with a natural number object.

        Earlier in the sequence on Wed, 14 Jan 1998 I said:

>>>You can do real analysis in any topos with natural numbers.

        Not to beat around the bush any longer let me give you a definitive
answer:

        You can do real analysis in any topos with a natural number object.
        
        This is plenty enough to build bridges. But you don't really mean
bridges. To repeat from the quote above:

> >   I mean the standard undergraduate material on
> >   ordinary and partial differential equations, power series, Fourier
> >   analysis, etc., complete with applications, examples, etc.

        But you do not literally mean the undergraduate curriculum at PSU
either. Kanovei Fri, 16 Jan 98 12:57:41 understood you correctly when he
wrote of:

>>>mathematicians working with *bridges* (which by Simpson means real
analysis >>>in its classical form)

        You want "real analysis in its classical form", the real analysis
you know and respect. And that is real analysis in ZFC. That is why you so
often ask about ZFC (I quoted those passages in my post of Fri, 16 Jan 1998
and you ignored them in your reply).

        To answer one more time:

        You can do real analysis in any topos with a natural number object.
It opens up a lot of interesting and useful possibilities beyond ZFC (of
course ZFC is already terrific itself) plus you can keep building bridges.

Colin