[FOM] Set Theory and Analysis

D.R. MacIver drm39 at cam.ac.uk
Wed Jan 12 09:34:36 EST 2005


This is a follow up to an email of mine a while ago about reverse 
mathematics and analysis. On that note, sorry to everyone who I didn't 
reply to (i.e. most of you). Completely my fault, and I promise to be much 
better this time. :)

Anyway, I've browsed the recommended book, subsystems of second order 
arithmetic, and intend to try and have a proper read through it when I have 
more time, but I decided that reverse mathematics isn't really the 
direction I want to take this.

I think I'm going to be focusing mostly on how the underlying set theoretic 
axioms influence analysis (and in particular I'm *not* going to be focusing 
so much on the implications of analyis in set theory proper)

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. (If anyone can point me 
to things you can do in analysis in ZF set theory but not in Z, I'd be very 
appreciative, but I strongly suspect there aren't really any. I certainly 
can't find even one). I'm not going to be looking at anything weaker than Z 
set theory.

So far I've got the following references: 

[1] Eric Schechter, Handbook of Analysis and its Foundations, Academic 
Press 1997 [2] H. G. Dales, W. H. Woodin, J. W. S. Cassels, An Introduction 
to Independence for Analysts, Cambridge University Press 1987 [3] Krzysztof 
Ciesielski, Set Theoretic Real Analysis, Journal of Applied Analysis 3(2) 
(1997), 143--190. (This is available online at 
http://www.math.wvu.edu/~kcies/prepF/56STA/STAsurvey.html ) [4] Shelah, On 
Ciesielski's Problems -- Journal of Applied Analysis 3 (1997), 191-209. 
(This is available online at http://shelah.logic.at/files/675.pdf )

I'm not sure about using [4] - most of the questions answered in it seem a 
bit too technical for the level I'm looking at. At the moment it's just 
included for completeness.

Anyway, thanks very much.

David MacIver

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