No subject

[Robert I. Soare]

TO:		FOM (Foundations of Mathematics)
FROM:	Robert Soare, University of Chicago
DATE:    Thursday, August 5, 1999
RE:		Priority Arguments in Reverse Mathematics
=========================================================================


I wrote on my web page and in (FOM,  2 Aug 1999 12:54:07) that

    > If one specifically desires applications of the priority method
    > to Reverse Mathematics, then Mytilinaios and Slaman used a
    > priority argument in their Reverse Mathematics paper, "On a
    > question of Brown and Simpson," and solved a problem posed by
    > Brown and Simpson in 1993.

When I wrote that paragraph, SLAMAN was LITERALLY standing over my
shoulder in my office at Berkeley and dictating it while I typed, so
that it would be EXACTLY correct.

Even so, Simpson (FOM, Aug. 3) claimed this paragraph was NOT correct.
>
>			Simpson (FOM, Aug. 3)
>
>     But this claim by Soare is incorrect.  I happen to be familiar with
>     this area!  The result in question is that BCT-II does not imply
>     RCA_0^+.  The Mytilinaios/Slaman proof of this result uses the low
>     basis theorem but does not use a priority argument.  (With regard
>     to the low basis theorem, see below.)  There is a priority argument
>     in the Mytilinaios/Slaman paper, but it doesn't address reverse
>     mathematics.

Slaman, upon seeing Simpson's message late Wednesday afternoon, Aug. 4, 
directly CONTRADICTED IT!   Slaman's reply can be found on the web site:

www.cs.uchicago.edu/~soare/lectures/slaman.text

In any case BOTH Slaman and Simpson agree that:  

(1) a priority argument occurs in the Mytlinaios-Slaman paper;

(2) Mytlinaios and Slaman solve a problem of Simpson and Brown of 1993.

(3) the problem is one in reverse mathematics.

----------------------------------------------------------------------

Therefore, there is clearly SOME intellectual connection in that
paper between the world of priority arguments and of reverse
mathematics.  Beyond that let's let Slaman and Simpson haggle
over the details.  I do not know them since I was not a co-author.


----------------------------------------------------------------------


NOTE.  I WAS a co-author of the LOW BASIS THEOREM which WAS used in
the Simpson's statement and proof above in Reverse Mathematics, and
which was DISCOVERED using a priority argument.  As a co-author I
strongly I maintain that the Low Basis Theorem, which is very often
used in applications, Simpson's book, and elsewhere) might NEVER have
been discovered without the priority method.


ANALOGY.  

Suppose Columbus discovers the New World, and fifty years later,
AFTER many round trips, Stephen Smith discovers a much BETTER trade
route, indeed the very BEST route from Spain to the New World, based
on maps, latitude, the appropriate great circle from Spain to the New
World, the winds, currents, etc., which he can actually PROVE (in
some base theory like RCA_0) is the absolute best.  Does Columbus
lose ALL intellectual claim to the ORIGINAL discovery?   Will
we now celebrate "Smith Say" in October every year?

This KEY point seems to be LOST on our MODERATOR, Stephen 
Smith (Opps, Simpson).

NOTICE.  According to the analogy of Columbus (I, with my co-captain
Jockusch) hereby CLAIM as INTELLECTUAL PROPERTY ANY and ALL
applications of the Low Basis Theorem as being indebted
intellectually, conceptually, practically, theoretically, and
mathematically to the PRIORITY METHOD.  Any person who employs,
refers to, mentions, or otherwise is inspired by the Low Basis
Theorem is HEREBY NOTIFIED he/she must pause to reflect on the
priority method for one moment of silence before proceeding with his
paper, research, book, or other scientific activity.

We have a Columbus Day.  Now we shall have a Priority Day to celebrate
and recall and give thanks for priority arguments.

I hope that in the future the Moderator and his chief program
chairman will,
>
>   have appropriate reverence and respect for the achievements and
>   possibilities of the priority method enterprise.  
>
>         Date: Thu, 10 Jun 1999 14:12:35 -0400 (EDT)
>         From: Stephen G Simpson <simpson at math.psu.edu>
>         Subject: FOM: awe; an unusual recursion theory meeting

	[Soare has taken some slight liberties in rephrasing this msg 
	to obtain the same meaning for PRIORITY ARGUMENTS which 
	Friedman and Simpson intended for the "f.o.m. enterprise."]

*************************************************************************
Robert I. Soare
Paul Snowden Russell 
Distinguished Service Professor
    of Mathematics and Computer Science			
Department of Mathematics 	     PHONE: (773) 702-6029, Secty: 702-7100 
The University of Chicago            FAX:   (773) 702-9787
5734 University Avenue		     E-MAIL:	soare at cs.uchicago.edu
Chicago, IL 60637-1546	USA  	     WEB: http://www.cs.uchicago.edu/~soare
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