FOM: miniaturization

[Jan Mycielski]

Dear Steve, 
	I have interpolated my answers (as JM) within your letter below
and erazed very little since all that you wrote is pertinent to my
question concerning the significance of Friedman's miniaturizations of
various set theories.
					Regards
					Jan

On Tue, 14 Sep 1999, Stephen G Simpson wrote:

> Dear Jan,
> 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.

JM: At this point I should add the following. My S(T), which in this
case says "every finite part of FIN(PA) has a finite model", is very
similar. The universe of that finite model corresponds to your {1,...,n},
But the coloring is much more special, so that my S(T) is weaker.
Actually it is equivalent to Con(PA) (while the Paris-Harrington statement
is stronger). To explain exactly my statement takes more space than the
P-H statement (exactly 17 lines), but it has the merit that it is once for
all, I mean for all theories not only PA and the merit of equivalence.
 
> 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).

JM: I believe in the unity of mathematics and mathematicians. Thus 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.
And 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.

> 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.

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 here
without the availability of subscripts and Greek letters would be too
ugly. Please consult JSL 51.
 
> 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).

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).

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

JM. I suppose I did. [But, I do not want to overstate the value of all
this! [Look at my previous fom letter where I try to put it in the right
perspective.]

> Best regards,
> -- Steve

Best regards,
	Jan