[FOM] Set Theory and Analysis
praatika@mappi.helsinki.fi
praatika at mappi.helsinki.fi
Wed Jan 12 11:13:50 EST 2005
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.
Best
Panu
Panu Raatikainen
Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki
Finland
Tel: +358-(0)9-191 23437
Mobile: +358-(0)40-840 0789
Fax: +358-(0)9-191 24509
Email: panu.raatikainen at helsinki.fi
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm
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