FOM: comments on Steel's "large cardinals needed/appreciated"

[Stephen G Simpson]

Referring to Feferman's "Does mathematics need new axioms?", Steel
says:
 > The talk begins by noting that the question is vague. I agree--the
 > crucial vagueness lies in "need". Unfortunately, I recall no
 > ensuing attempt to sharpen the question.

Feferman's paper discusses several specific sharpenings of the
question.  Among them are (1) which axioms are useful for scientific
applications?  (2) which new axioms are likely to play a role in
settling basic set-theoretical questions such as the continuum
hypothesis?  (I'm not saying that Feferman's discussion is anywhere
near the last word on these sharpened questions; as I have pointed
out, Feferman omits some relevant evidence.)

 > He argues that all scientifically applicable mathematics can be
 > developed in a system W which is conservative over PA, and thus at
 > the moment we need no more than W (for science? for mathematics?).

For science, obviously.  

 > does this mean we should stop developing the mathematics outside W?

I don't think Feferman ever suggested this.  But it's an interesting
suggestion.  :-) 

Seriously, there always has been and always will be a distinction
between scientifically applicable mathematics and pure mathematics,
i.e. mathematics pursued for its own sake, though obviously these two
overlap.  If scientifically applicable mathematics requires only
relatively weak axioms, but pure mathematics requires stronger axioms,
then why not let the pure mathematicians go their own way?  Live and
let live.

 > It would require a massive re-education project to teach humanity
 > to live within W.

Are you sure?  Right now, I imagine that a typical undergraduate
course in foundations of mathematics would consist of naive set theory
followed by the Dedekind/Cauchy construction of the number systems
within set theory, etc.  It might be an interesting experiment to
replace naive set theory with a naive version of W or of some other
relatively weak system.  Would this be so very difficult?

 > if we are not going to try to live within W, what is its practical
 > relevance to mathematical behavior?

This strikes me as a rather short-sighted question.  But I'll try to
answer it anyway.

Practical relevance?  It depends on what you mean by practical.

To me it's incredibly intriguing that the applicable parts of
mathematics seem to need only relatively weak axioms.  This strikes me
as a mathematical and scientific insight of the utmost importance.
One can argue that this insight ought to be incorporated into the
undergraduate curriculum.  On a more lofty level, this same insight
may well be a significant clue regarding the nature of mathematics and
its relationship to the rest of human knowledge.  There may be various
ways to pursue this and make it rigorous.  My tentative explanation is
that the commonly considered weak axiom systems (PA and weaker) tend
to focus on concrete mathematical objects such as the natural numbers
which have some sort of real world existence, while the commonly
considered strong systems (ZFC and stronger) tend to focus on things
like aleph_omega and large cardinals, whose real world existence is
dubious at best.  This may account for the applicability of weak
axioms.  Can we somehow formalize these questions and insights,
perhaps in terms of axioms for physics, or perhaps axioms for a more
general theory of real world entities and processes?  This could lead
to major breakthroughs in f.o.m. and foundational studies generally.

Speaking of the need for large cardinal axioms, Steel says:
 > One does seek applications in less theoretical areas, as close to
 > central and active topics as one can find them.  In the
 > case of large cardinal axioms, one has the descriptive set theory
 > of projective sets of reals--and beyond--which they yield.

Projective sets?  If you are going to talk about topics as close as
possible to central mathematics, why not mention Borel and analytic
sets?  And finite combinatorial mathematics?  These topics are much
closer to core mathematics than projective sets.

Regarding Borel and analytic sets, even Lou van den Dries, the
hard-nose Bourbachiste, agrees that they are interesting.  And people
like Baire, Borel, Lebesgue, Souslin, and Lusin would have agreed; I
recently reread chapter IV of Fraenkel/Bar-Hillel/Levy regarding the
French "semi-intuitionists".  And there are a number of results
showing that higher cardinals are needed.  I'm thinking of Friedman's
results showing that aleph_1 cardinals and in some cases large
cardinals are needed to prove natural statements about Borel
functions.  There is also a result of Steel saying that sharps are
needed to prove that any two analytic non-Borel sets are Borel
isomorphic.  And I think Kechris and Hjorth have also had something to
say about this.

By contrast, some people might view the results about projective sets
as fairly empty generalizations of well known results about analytic
and coanalytic sets.

In the realm of finite combinatorics, there are Friedman's results
providing fairly natural combinatorial statements that require large
cardinals (Mahlo cardinals, subtle cardinals, etc) to prove.  Friedman
thinks that these combinatorial statements will grow into significant
branches of mathematics.  The jury is out on this.  Granted, the large
cardinals involved tend to be not as large as the ones that have been
used to prove theorems about projective sets.  But so what?

I agree with Steel on one point.  I would put it this way: An
unfortunate aspect of Feferman's article is that he omits a lot of
interesting developments which tend to show the need for new axioms,
in various senses of the word "need".  In this sense, Feferman's
article may act as a kind of gratuitious wet blanket, unnecessarily
dampening enthusiasm for this kind of research, which is vital for
f.o.m.

Sincerely,
-- Steve