[FOM] Set Theory and Analysis
D.R. MacIver
drm39 at cam.ac.uk
Wed Jan 12 12:13:32 EST 2005
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
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