[FOM] Set Theory and Analysis

[D.R. MacIver]

On Jan 12 2005, praatika at mappi.helsinki.fi wrote:

> D.R. MacIver" <drm39 at cam.ac.uk> wrote:
> 
> > What I'd really like are some references to books and papers on this 
> > subject: Both on questions in analysis which are independent of ZFC, 
> > and how much analysis one can do in theories weaker than ZFC. In 
> > particular what happens when you weaken the axiom of choice.
> 
> All the ordinary analysis can be developed even in ACA_0, which is 
> conservative over Peano Arithmetic PA. (see Simpson's book; the 
> obserevation goes back, I think, to Friedman 1976, Feferman 1977 and 
> Takeuti 1978)
> 
> In terms of set theory: take ZFC without the axiom of infinity, and add 
> the negation of the latter. Call the resulting finitary set theory F. PA 
> and F are not only relatively interpretable in each other, but even 
> 'logically synonymous' (Visser 2004), and thus equivalent in a very 
> strong sense. Then add to F the comprehension scheme exactly as you 
> extend ZFC to get GB. The resulting theory is conservative over F, and is 
> the set theoretical counterpart of ACA_0. Thus one can develope all the 
> ordinary analysis in this theory.
> 

This depends very strongly on precisely what you mean by `ordinary 
analysis'. It is my impression that this restricts you to seperable spaces 
for many theorems, and makes a lot of functional analysis impossible or 
extremely difficult. I dont know enough about the subject to say for sure 
though, so I might be wrong.

Regardless of which, it is also not the direction in which I want to take 
the essay. I'm talking purely about analysis in theories containing Z (and 
almost exclusively ZF as well) set theory.

David