FOM: miniaturization

[Stephen G Simpson]

Dear Jan,

I am glad to see you here!  I know that you have been wrestling with
the question of ``To FOM Or Not To FOM'' for a long time.  I am glad
you finally decided to FOM.

You wrote:

 > the statements of Paris and Harrington (for PA), and those of Friedman
 > which I saw, do not appear to me more appealing than S(T) ...

I propose that we try to get to the bottom of this issue right here on
the FOM list, by simply comparing the corresponding statements side by
side.

First, let's look at finitary statements that imply Con(PA).

The Paris-Harrington statement is:

  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
  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.

Let's call this statement P-H.  I think we can agree that P-H is
reasonably natural and appealing from the mathematician's point of
view.  (The kind of mathematician I have in mind is a finite
combinatorist, a graph theorist or somebody like that.  When I say
mathematician, I am emphatically *not* talking about logicians.  If we
were talking about logicians, we could simply say Con(PA) and be done
with the whole issue.)  Specifically, P-H closely resembles the finite
Ramsey theorem.  Indeed, it is the same as the finite Ramsey theorem
except for the last clause, card(X) > min(X).

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.

By the way, in addition to P-H we could also compare S(PA) to some
more recent statements of Friedman which also imply Con(PA) and are
even more mathematically natural than P-H.

Then later, after we have gone through this comparison of statements
that imply Con(PA), I propose that we move on and look at statements
that imply Con(ZFC).  I am not sure which is Friedman's latest and
greatest finitary statment in this vein, but let's ask him to spell it
out here on FOM list, and then you can spell out S(ZFC) and we can
compare and contrast.

Actually I think that Friedman's current statements are stronger in
that they imply things like Con(ZFC + Mahlo cardinals) or maybe
Con(ZFC + subtle cardinals).  But I don't think this will make much
difference to the issue that you raise.  Let's just compare them to
S(ZFC).

Jan, what do you say?  Do you accept this challenge?

Best regards,
-- Steve