FOM: topos theory

[Kanovei]

It follows from McLarty's post that pumping 
enough ZFC into a topos we get inside the 
topos "reals" adequate for this or that 
purpose, for instance those not satisfying 
the excluded middle. 

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 ? 

Hopefully McLarty could address this question. 

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. 

Vladimir Kanovei