FOM: miniaturization

[Stephen G Simpson]

Dear Jan,

First I'll reply to some of your individual points, then I'll comment
on the general issue that you have raised.

Jan Mycielski 15 Sep 1999 16:38:37 writes:

 > the separation between those who know enough logic to undestand
 > Con(T) and those who do not seems too subjective to motivate any
 > mathematical work.

If the distinction between Con(T) and other mathematical statments
cannot motivate mathematical work, then what is the motivation for
your work on S(T)?  I know that your work is somehow motivated by
finitism, but aren't Con(T) and S(T) equally finitistic?

The first question above is rhetorical or ironical.  The reality is
that there is a large ``understandability gap'', because core
mathematicians and other scientists do not understand Con(T) except in
the relatively rare cases where they have studied mathematical logic
up through G"odel's incompleteness theorem.  And even when they
understand Con(T), there is still an ``appreciation gap'', because
they do not consider it to be ``natural'' from the viewpoint of core
mathematics.

So, we logicians have a problem.  The problem is that, while we
logicians appreciate the crucial importance and general intellectual
interest of statements like Con(PA) and Con(ZFC), there is no obvious
way to convey this appreciation to our colleagues in core math and
other scientific disciplines.

I believe your work on S(T) and the work of Paris-Harrington and
Friedman on finite combinatorial independence results are both
motivated in part by the twin problems of the ``understandability
gap'' and the ``appreciation gap''.  But they approach the problem in
very different ways and with very different outcomes.  There is no a
priori reason to think that either approach should replace or
supercede the other.  I mention this now, because in your original
``miniaturization'' posting of September 14, you seemed to suggest
that your work ought to render that of Paris-Harrington and Friedman
obsolete or irrelevant.

In order to evaluate your suggestion, I spelled out the
Paris-Harrington statement in complete detail, and then I asked you:

 > > Now, what is your statement S(PA) exactly?  After you spell out
 > > S(PA) in complete detail here on the FOM list, we can judge
 > > whether it is as mathematically natural and appealing as P-H.

You replied:

 > JM: As told above I did it in JSL 51 (1986), pp. 59 - 60, and it
 > took only 17 lines (for any T, and not only for PA). But copying
 > those lines her without the availability of subscripts and Greek
 > letters would be too ugly. Please consult JSL 51.

OK, I will consult JSL 51 for the precise statement of S(PA).  But
even before consulting JSL 51, your description of S(PA) seems to
indicate that S(PA) must be mathematically much more cumbersome and
less appealing than P-H.  If S(PA) is 17 lines long and cannot be
written without the use of subscripts and Greek letters, then that
sounds ugly indeed!  By contrast, P-H is only 5 lines long and needs
only ordinary Roman letters.  :-)

Incidentally, my earlier statement of P-H contained a typographical
error.  Here it is again, with the typo corrected.

  For all k, l, m there exists n so large that, if you color the
  k-element subsets of {1,...,n} with l colors, then there will be a
  subset X of cardinality at least m all of whose k-elements subsets
  have the same color, and such that the cardinality of X is greater
  than the smallest element of X.

See?  No Greek letters, no subscripts, 5 lines, mathematically
appealing.

Incidentally, Harvey has some statements that imply Con(PA) yet are
even more mathematically appealing than P-H from some points of view.
One such set of statements comes out of the Kruskal stuff.  For the
most recent developments in this vein, see Harvey's FOM posting of 20
Oct 1998 10:13:42.

By the way, when you say that S(PA) is 17 lines long, does that
include the statement of the axioms of PA?  If not, then we had better
add those axioms, in order to make the statement of S(PA)
mathematically self-contained, like the statement of P-H above.  So
now we could be up to around 25 lines.  Is this correct?

I was hoping I could get you to take your best shot at writing S(PA)
here on the FOM list, so that we can compare and contrast it to P-H.
But now I am beginning to see why you don't want to do that.

 > JM: If you do it once in my way you have it for any theory T and
 > moreover the statement is equivalent to Con(T).

Yes.  S(T) is defined uniformly in terms of T, and it is provably
equivalent to Con(T) over some weak base theory.  But Con(T) itself
also has these same nice features.  How is S(T) better than Con(T)?
Is this question adequately answered in JSL 51 or Lavine's book?  I
will have a look ....

Also, if S(T) can be explained only in terms of the axioms of T, then
that would seem to automatically move S(T) away from mathematical
naturalness.  For instance, the axioms of ZFC are not directly
connected to the normal working context of core mathematicians.

Now let's get back to the larger issue that you raised: What is the
point of P-H and Friedman's statements?  Or, as you put it:

 > Could somebody explain to me why H. Friedman is building various
 > statements of finite (or infinite) combinatorics equiconsistent
 > with various large cardinal axioms?

I will answer your question this way.  Friedman's goal is to find
necessary uses of strong axioms for deriving finite/discrete
mathematical statements which are ``natural'' according to the current
standards of naturalness employed by present-day ``working
mathematicians''.  And when he says ``working mathematicians'' he
definitely does not mean logicians.  He means core mathematicians.  Or
maybe mathematicians working in finite/discrete mathematics: Graph
theorists, combinatorists, people like that.

You said:

 > I know that you do not mean that Harvey's work in this area is
 > meant only for those who do not know enough logic.

Yes, that is correct.  One of Harvey's heuristic goals is to make his
statements as appealing as possible to mathematicians who know no
logic.  Presenting Harvey's statements to mathematicians who know no
logic is a good test of the fundamental point made by his work in this
area.

Would presenting S(T) to people to know no logic be a good test of the
fundamental point made by your work on S(T)?

If you think about this last question, I think it will become clear to
you that the goals and outcomes of Friedman's work are quite different
from the goals and outcomes of your work on pages 59-62 of JSL 51.

Best regards,
-- Steve