FOM: topos theory

[Kanovei]

>Date: Fri, 16 Jan 1998 14:50:54 -0500 (EST)
>From: cxm7 at po.cwru.edu (Colin McLarty)

>You want "real analysis in its classical form", the real analysis
>you know and respect. And that is real analysis in ZFC.

This statement needs to be commented. 

The foundation and development of the 
"real analysis in its classical form" is 
mostly associated with the activity of 
Cauchy, Dedekind, Weierstrass, Borel, Baire, 
Lebesgue (of course many names are missing, 
but I mean the ideas rather than the names), 
to which we may add the subsequent development 
of probability, measure theory, and perhaps 
descriptive set theory (first known as the 
descriptive theory of functions). 

I think it is clear that all the above was 
created in the full explicit ignorance of ZFC. 
In this sense the classical real analysis is 
definitely NOT the *real analysis in ZFC*. 

The logical and actual sequence of events was just opposite:

set theory (including ZFC) accumulated 
some ways of mathematical reasoning from real analysis. 
This is why the real analysis so nicely fits to ZFC 
that a modern category theorist may be about to think that 
the classical r.a. is just a part of the general ZFC plot. 

In this connection, let me repeat a question of 
Steve Simpson which has not yet been answered and which 
he may have been tired to ask once again. 

QUESTION. We know that taking a topos with SOMETHING 
is enough to develop more or less full version of the 
classical real analysis. Now, can this SOMETHING be 
f.o.m.-motivated along the general lines of motivation 
in the category theory ?

Possible answers: 

1) yes, and the motivation is ....

2) no, the SOMETHING is just a more or less explicit 
   part of ZFC

3) we do not care much about the classical real analysis 
   as it is not enough algebraic to be in "core mathematics". 

4) as the category theory pretends to embrase the mathematics 
   in its full totality, employing ZFC as SOMETHING is 
   pretty much allright with us.

Vladimir Kanovei