FOM: topos theory

[Kanovei]

>From: "Michael Thayer" <mthayer at ix.netcom.com>
>Date: Fri, 16 Jan 1998 08:11:02 -0600

>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 have no problem with that. I wrote that the 
topos theory way to ground real analysis is not 
inspiring -- in the sense that it does not inspire me. 

But really, how it can ? 

I am working sometimes with Borel sets of the real 
line. Being aware of some paradoxes, I feel need in 
some background, to avoid wrongdoings in my study. 
Such a general setup, now called ZFC, has been created 
and developed to the extent that 
1) it perfectly fits to my intuition of the mathematical 
   universe, in particular, Borel sets in R, 
2) once accepted, it allows almost not to think about 
   foundational problems in my (concrete) study of 
   Borel sets in R. 

Now a category theorist comes and says: this everything 
is not true, the real picture is that THE 
real numbers do not exist while you can get something 
he calls reals in every such-and-such topos, and this 
something in general depends on which topos I take, so 
what then be my theorems about ? 

To conclude I would consider to change the set theoretic 
setup for my study in mathematics to category theoretic 
(or any else) if I had seen a list of preferences from 
the change (in the most wide sense of the word: preference) 
-- for my study of Borel sets -- 
which weights more than the list of losses. At the moment 
the list of preferences is empty while the list of losses 
is essentially non-empty (see above). This is why I wrote that 
I was not inspired. 

>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 not an expert on KF, so I would refrain from 
answering your direct question. 

Vladimir Kanovei