[FOM] Set Theory and Analysis
D.R. MacIver
drm39 at cam.ac.uk
Wed Jan 12 09:34:36 EST 2005
Hi,
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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