[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. 



Panu Raatikainen

Helsinki Collegium for Advanced Studies
P.O. Box 4
FIN-00014 University of Helsinki

Tel:  +358-(0)9-191 23437
Mobile:  +358-(0)40-840 0789
Fax:  +358-(0)9-191 24509
Email: panu.raatikainen at helsinki.fi


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