Re: Cauchy completeness of Cauchy reals

[Toby Bartels <toby+categories@math.ucr.edu>]

Dusko Pavlovic wrote in part:

>(i think this does deserve some general interest because we often say
>that categories capture real mathematical practices --- but as it
>happens, cauchy reals are not complete, and the mean value theorem
>fails, and so on. i was hoping to understand where does the usual
>intuition of continuum fail, and what categorical property do we need
>to do basic calculus.

It depends on what you mean by "basic calculus".
Bishop would argue that he can do basic calculus just fine
using a constructive version of the mean value theorem.

This is not to say that you don't have an interesting question;
from the POV of the mathematician on the street (not very theoretical),
classical theorems often follow from constructivist (a la Bishop) one
if you assume that sequentially compact metric spaces are compact
(which means complete and totally bounded to Brouwer and Bishop),
so that might be one place to look.


-- Toby