FOM: set-theory-1 versus set-theory-2

[Stephen G Simpson]

Shoenfield 12 Sep 1998 15:31:43 has presented a concise overview of a
certain line of research in set theory.  I think his account is fairly
standard in the sense that it summarizes how most leading
set-theorists view the subject.  However, there is another
perspective.

First, a preliminary remark.  When discussing set theory, there is an
important distinction to be made.  Let me formulate this as a division
of set theory into two distinct subjects, set-theory-1 and
set-theory-2.  (Others may want to view these not as two subjects but
as two perspectives on the same subject.  I don't think this will
matter very much.  What matters is that we recognize the distinction
between set-theory-1 and set-theory-2 in some form or another.)

Set-theory-1 is a particular specialty within mathematics, focusing on
the study of sets of various kinds, uncountable cardinals, etc., both
for their own sake and for applications and connections to other
branches of mathematics such as analysis, algebra, topology, etc.
This kind of set theory is a substantial and respected branch of
mathematics, perhaps 1 or 2 percent of mathematics as measured by
number of articles published, etc.

Set-theory-2 is set theory viewed as a foundation for *all* of
mathematics -- not only *a* foundation, but in fact *the orthodox* or
commonly accepted foundation for *all* of mathematics, in the present
historical era.  This aspect of set theory is crucial for philosophy
of mathematics and f.o.m.

Admittedly set-theory-1 and set-theory-2 are related and there is much
overlap.  Nevertheless, I would insist that there is a need for the
distinction, because set-theory-2 is absolutely crucial for f.o.m.,
while in a very real sense, set-theory-1 is arguably not more
important for f.o.m. than any other branch of mathematics.

Although it's seldom discussed explicitly, I think this distinction is
well recognized, especially by outsiders, i.e. non-mathematicians and
other non-set-theorists who are interested in set theory.  The
distinction has been aired on the FOM list back in July in connection
with the Mathematical Subject Classification, 04XX versus 03EXX; see
for instance my posting of 8 Jul 1998 17:54:51.

Shoenfield briefly alludes to the distinction:

 > A conclusion of this chapter is that virtually all of
 > accepted mathematics can be formulated and derived in ZFC.  This
 > conclusion has many applications, but it does not, in my opinion,
 > shed much light upon the nature of set theory.

This remark is somewhat opaque, but I understand it as follows:
set-theory-2 has a lot of general intellectual interest, but it does
not shed much light on set-theory-1.  This reading of Shoenfield's
remark is partially confirmed by the fact that the rest of
Shoenfield's discussion is concerned almost exclusively with
set-theory-1.

Furthermore, from the perspective of set-theory-1, Shoenfield's
account is balanced and makes a lot of sense.  Cantor's continuum
problem and the various classical problems concerning regularity
properties of projective sets, etc., are indeed central problems of
set-theory-1, and the progress on these problems has been substantial
and impressive.  Moreover, an exciting feature which sets set-theory-1
apart from other branches of mathematics has been the evident pressing
need for new axioms.  From this point of view, the Martin-Steel
theorem is indeed a high point of set-theory-1, though surely not the
end of the unfolding story.

However, I would like to point out that all of this looks quite
different from the perspective of set-theory-2, i.e. set theory qua
foundations of mathematics as a whole.

Here's the key point: From the perspective of f.o.m. and mathematics
as a whole, the problems dealt with in set-theory-1 (the continuum
problem, regularity properties of projective sets, etc.), are *not*
central mathematical problems.  They are almost exclusively the
concern of set-theorists, i.e. specialists in set-theory-1.

It's true that some tools of set-theory-1 have occasionally caught the
attention of non-set-theorists.  Examples: the use of AC and CH to
construct pathological sets of reals giving counterexamples in
analysis; Shelah's work on the Whitehead conjecture concerning
uncountable Abelian groups; etc. etc.  However, the vast majority of
mathematicians justifiably regard the techniques and concerns of
set-theory-1 as almost entirely irrelevant to their own specialties.
In branches of mathematics other than set-theory-1, plenty of progress
can be made without new axioms, and the need for new axioms has been
marginal or barely recognized or not recognized at all.  Moreover,
even when there are serious obstacles to progress in such branches of
mathematics, the new axioms that play such a key role in set-theory-1
do not seem likely to be of any use.  There is even a metamathematical
explanation of why this is so: *absoluteness*.  The Shoenfield
absoluteness theorem says that forcing does not affect the truth value
of Sigma^1_2 or Pi^1_2 sentences.  Most mathematical questions have
equivalent formulations which lie well below this level of the
familiar hierarchy of formulas in the language of second order
arithmetic, so forcing is irrelevant to such questions.

The result is that mathematicians outside set-theory-1 tend to be
justifiably suspicious of the usefulness and importance of new axioms.
Such suspicions are heightened by the sometimes confusing relationship
between set-theory-1 and set-theory-2.  It is understandable if the
majority of mathematicians, who are not set-theorists, get the
impression that independence and new axioms are important only for
f.o.m., not for what they call "core mathematics".

Returning now to the perspective of set-theory-2 or f.o.m., it seems
very natural to go back and reexamine the set-theory-1 program of
independence results and new axioms, this time with an eye to
extending these phenomena into "core mathematics", beyond the
specialized branch of mathematics known as set-theory-1.  An important
goal here would be to find statements of a more mathematical, less
set-theoretic character that are nevertheless independent of ZFC or
other foundationally interesting axiom schemes.  One of the criteria
for success would be, how close are these independent statements to
the central concerns of various mathematical specialties, other than
set-theory-1.  This is the program of "mathematical incompletness"
which has been pursued most vigorously by Harvey Friedman.

I would urge Harvey to say more about this.  Harvey's posting
"background on incompleteness I" (5 Sep 1998) is a start.  I am
looking forward to parts II, III, ....

That's basically all I want to say about this right now.  However, let
me make two further comments on Shoenfield's account of set theory.

Shoenfield says:

 > What impresses me in the whole story is how the solution of
 > problems leads to new concepts, which are then developed and, after
 > a time, integrated with the old concepts.

This aspect is not particularly impressive to me, or perhaps I should
say, it is not a distinguishing feature of set theory.  Most
mathematical and non-mathematical subjects exhibit this feature.

 > To me, this shows the folly of trying to decide in advance of doing
 > the mathematics what the fundamental concepts of set theory are.

My first reaction to this statement is negative.  To me it sounds like
a veiled attack on the foundational or f.o.m. perspective.  What does
it really mean?  Does it mean that Cantor, Frege, Russell, von
Neumann, et al were exhibiting folly when they developed the
fundamental concepts of set theory?  I don't think so, but it's
important to be as clear as possible about these things, and in my
opinion Shoenfield's formulation is not nearly clear enough.

Perhaps Shoenfield is merely pointing out that, historically,
important concepts emerged in advanced stages of the field.  If so,
that I would say that is true but unsurprising.  It's certainly not a
valid argument agains the foundational, as opposed to the
mathematical, perspective.

-- Steve