[FOM] Fwd: Re: analysis and set theory
D.R. MacIver
drm39 at cam.ac.uk
Tue Jan 18 13:18:10 EST 2005
Just forwarding (with permission) a more complete explanation of the types
of theorems in analysis which depend sensitively on the underlying set
theoretic axioms that I sent to Randall Holmes. This should clarify my
rather brief response to the email list as to why second order arithmetic
isn't really enough for the kinds of analysis I'm interested in dealing
with.
I've chopped out the original email this is replying to, as well as various
non-mathematical bits.
---------- Forwarded message ----------
From: "D.R. MacIver" <drm39 at cam.ac.uk>
To: Randall Holmes <holmes at diamond.boisestate.edu>
Subject: Re: analysis and set theory
Date: 12 Jan 2005 19:05:00 +0000
<stuff snipped>
Two major examples are the study of measure theory and automatic continuity
in banach algebras - both of these are affected by the presence of the
continuum hypothesis.
For example, consider a function f : [0, 1]^2 -> [0, 1]. Fubini's theorem
says that whenever this function is measurable then the two iterated
integrals Int ( Int f(x, y) dx) dy and Int ( Int f(x, y) dy ) dx are equal.
Does a generalised version of this hold? i.e. whenever the functions x ->
f(x, y) and y -> f(x, y) are measurable (for fixed x,y respectively) and
the iterated integrals exist, are they equal? The answer proves to be
independent of ZFC, but CH (indeed MA is enough) settles this question with
the answer no.
Automatic continuity is a more subtle area. The fundamental question is the
following: Let X be a conpact hausdorff space, C(X) the banach algebra of
continuous complex valued functions on X. Let B be another banach algebra.
Is every homomorphism C(X) -> B continuous. Again, the answer is
independent of ZFC. (And again, CH answers the question negatively).
See the references I provided in the FOM mailing list (particularly Woodin
and Dales, and Cierpinski) for more details of the above.
Incidentally, as to the question of whether choice makes a difference, the
primary concern is how weakened forms of choice make a difference. For
example if we wanted to instead work in ZF + AD, we get some rather nice
consequences (dependent choice works, which is enough for a good deal of
classical and functional analysis, but all subsets of R are measurable).
<more stuff snipped>
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