# FOM: MSC abolishes Set Theory !!!

Stephen G Simpson simpson at math.psu.edu
Wed Jul 8 17:54:51 EDT 1998

```Thomas Forster deplores

> the viewpoint that Set Theory is part of foundations and it is this
> view, i think, which causes a lot of the misunderstanding of Set
> Theory (more prevalent over this side of the pond, admittedly) and
> the hostility we experience.  A lot of mathematicians think either:
> set theory is foundations and who cares about foundations? or: set
> theorists think it's their job to tell us the meaning of what we
> do.

Interesting.  If I may paraphrase, you are saying that you want to
distance yourself from f.o.m. (= foundations of mathematics) as much
as possible, because in your experience set theory qua f.o.m. leads to
misunderstandings and attracts hostility from many mathematicians.

What forms has this hostility taken?  Would you care to tell us some
of your favorite "tales from the front"?

I myself reject this kind of hostility.  I care deeply about f.o.m.,
and I do think it's the job of f.o.m. to explain the meaning of
mathematics.  So I prefer to confront these hostile mathematicians
head on, rather than meekly surrender to them or try to get along with
them, as Forster seems willing to do.

About the MSC issue (04XX versus 03EXX), perhaps we could say that set
theory has a dual nature, or there are two aspects of set theory.  Let
me try to formulate these two aspects.

On the one hand, set theory may be viewed as a branch of mathematics
just like any other, and it can be studied for its own sake or in
terms of its connections to other branches of mathematics such as
measure theory and topology; this is the viewpoint of Hausdorff's book
Mengenlehre, and of people like Erdos, Rado, and Hajnal, who do
infinitary combinatorics as a generalization of finitary
combinatorics.  From this point of view, set theory is a respectable
but narrow niche; it is probably not of much interest to most
mathematicians, and it certainly does not have much in the way of
general intellectual interest.

Set theory students: Is this the kind of field that you want to
specialize in?

On the other hand, there is the viewpoint that set theory is *not*
just another branch of mathematics.  Rather, set theory has a special
significance as a very successful foundation for *all* of mathematics;
this is the view of G"odel, Cohen, Friedman, ....  From this point of
view, set theory has tremendous general intellectual interest and is
potentially of great significance to all mathematicians.  This is
obviously a much loftier and more ambitious view of set theory.

Set theorists, how do you view your subject?  04XX or 03EXX?

-- Steve

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