FOM: MSC abolishes Set Theory !!!

[Stephen G Simpson]

Alex V. Kuz 08 Jul 1998 08:10:19 writes:

 > Sets are very interesting objects (especcially objects(toposes)
 > modelling sets in various categories) and certainly set theory is a
 > subject on its own !

Yes, set theory is a subject.  But the question is whether this
subject should be considered as independent of (i.e. disjoint from)
another subject, logic and foundations.  The history of set theory
seems to suggest that it is closely connected with logic and
foundations.

A set theorist has told me off-the-record that several set theorists
advocated abolition of MSC 04 for pragmatic reasons:

  1. There was a lot of duplication between the subcategories of 03E
  (logic and foundations -- set theory) and 04 (set theory)
  respectively.  This led to a lot of confusion.
  
  2. For the past several years, serious set theorists have tended to
  classify their own publications as 03E in preference to 04.  The 04
  classification was populated mostly by articles on fuzzy set theory,
  etc.

According to my set-theoretic informant, these pragmatic reasons have
nothing to do with the intellectual issue of whether set theory is to
be considered as part of f.o.m. or not.

Does anybody know how the 03E and 04 classifications came about in the
first place?

Joe Shipman 8 Jul 1998 14:36:37 writes:

 > I think the reason the classification was modified to put set
 > theory "under" foundations is that it is impossible to work in set
 > theory without constantly running into foundational issues, because
 > independent statements are everywhere.

Intellectually, this seems like a good reason to put set theory under
foundations, i.e. to merge 04 into 03E.  But from what I have been
able to find out, the actual reasons for the merger were more
pragmatic.  From what I have been able to find out, set theorists
don't seem to care much about the distinction between (i) set theory
qua specialized branch of mathematics versus (ii) set theory qua
f.o.m.  If I'm wrong about this, would some set theorists please
correct me?

 > This seems to me a mistaken decision, though not an intolerable
 > one.  

Why do you think it's mistaken?

 > It would become intolerable if any of the following developments
 > takes place: 1) An alternative non-set-theoretic foundation of
 > mathematics is found that "works" at least as well as the standard
 > set-theoretic one.  2) The current set-theoretic foundations are
 > augmented by the general acceptance of higher axioms such as very
 > large cardinals, which will make set theory a more "ordinary"
 > subject where one can simply claim to have proved "X" rather than
 > "ZFC plus axiom A implies X".  3) Friedman's program of showing
 > that independent statements are everywhere in non-set-theoretic
 > mathematics as well succeeds.

Yes, but note that all of these projected developments would be
revolutionary.  They would require huge advances in f.o.m. research,
and they would massively enhance the prestige of f.o.m. within the
mathematical community.

-- Steve