FOM: Martin-Steel theorem
jrs at math.duke.edu
Tue Sep 8 15:27:22 EDT 1998
This is a brief reply to two recent postings by Steve.
I regret that in my possible conjecture, all of the 1's should have
been 0's. By a strictly pi-0-n+1 sentence I mean one which is not
pi-0-n or sigma-0-n. I stated this conjecture only because you
demanded one, but I would really rather return to my original general
statement: one should look for a result which relates the position of an
undecidable statement in the arithmetical or analytic hierarchy and the
number and kind of large cardinals needed to prove it.
My statement about what has made and will make a splash did not
refer to any programs, but to two particular theorems which have been
proved. I have always been puzzled as to why you considered the par-
ticular result of Harvey such a key result in the completeness program.
I agree that it would not be useful for us to plunge into a debate
on the relative virtues of the two programs. I find your challenge to
explain why I value the Steel-Martin theorem so highly not only
reasonable but welcome, since it will afford me a chance to express
some thoughts on judging mathematical results which I have had recently.
I hope to meet the challenge sometime soon.
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