[FOM] set theory/CH

Monroe Eskew meskew at math.uci.edu
Sat Aug 23 02:05:58 EDT 2014

On Aug 21, 2014, at 6:49 PM, Harvey Friedman <hmflogic at gmail.com> wrote:

> I think I should have distinguished between descriptive set theory and "higher" set theory. The former is going to be around for longer than the latter. A sociological symptom that there is something wrong can be seen by observing the almost total lack of interest in not only in "higher" set theory but mathematical problems with any overtly "higher" set theoretic content. (Mathematicians are willing to use limited doses of "higher" set theoretic content in the service of proving theorems without such content, but although this happens, it is quite unusual). I would be interested to hear about any exceptions among the non logicians at the so called leading math depts - Stanford, Berkeley, Chicago, MIT, Harvard, Yale, Columbia, Princeton, IAS, etc. Of course, that's not the actual phenomenon, that's just the sociological symptom. 
> As I said, there is significantly more interest in descriptive set theory than higher set theory, but the interest even in descriptive set theory is dangerously low. 
> If you think this is false, or even if you just think that it may be true, but ought not to be true, why don't you report on a bunch of developments that you find "striking" or "important" or whatever, and the FOM can have a discussion. A possible outcome might be that people like me find that there is more interest in such things than we thought, but I think it likely that it is still very limited compared to the size of the math community.

I am not sure where “higher” set theory begins, and I cannot speak to the issue of “leading math departments.”  But I can give a few examples of the interaction of set theory with “ordinary mathematics” that seem interesting.  I hope others who are subscribed to this list can chime in with more information.

First is the very active work on operator algebras using set-theoretic methods.  Ilijas Farah spoke about this at the ICM meeting in Korea.  It is not something I know much about, but here are some key points from Farah’s paper (http://www.math.yorku.ca/~ifarah/Ftp/icm9.pdf).

(1) A a number of long-standing problems about C^* algebras have been solved or proved independent by set-theoretic methods.
(2) The methods are quite diverse and include descriptive set theory, model theory, cardinal arithmetic, and forcing axioms.
(3) In particular, Philips-Weaver and Farah showed that the question of whether there are non-inner automorphisms of the Calkin algebra— the space of bounded linear operators on the separable Hilbert space modded out by the compact operators— is independent of ZFC.

Another example which is closer to my own work is the use of “higher” set theory to solve some natural questions about measures on the real line.  These are ZFC theorems, but the proofs use large-cardinal-like methods.  The first addresses an old question of Ulam.

Theorem 1 (Gitik-Shelah): For every collection of countably many 0-1 valued nontrivial measures on R, there is a subset of R that is not measurable with respect to any of these measures.

(The measures are assumed to be countably additive, take values only 0 or 1, and nontrivial means all singletons are null.)

Theorem 2 (Kumar): For every set of reals A, there is a subset B with the same Lebesgue outer measure, such that the distance between any two members of B is irrational.

Kumar’s theorem is a nontrivial application of a nontrivial result of Gitik-Shelah from the same project in which Theorem 1 was proved.
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