FOM: Martin-Steel theorem

Joseph Shoenfield jrs at
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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