FOM: Martin-Steel theorem

Stephen G Simpson simpson at
Wed Sep 9 13:18:12 EDT 1998

Joseph Shoenfield writes:

 > By a strictly pi-0-n+1 sentence I mean one which is not pi-0-n or
 > sigma-0-n.

I still don't understand.  By adding dummy quantifiers, it is trivial
to convert a Pi^0_1 sentence to a logically equivalent Pi^0_{n+1}
sentence which is not Pi^0_n or Sigma^0_n.

You could avoid this difficulty by talking about sentences up to
equivalence over ZFC or something of the sort, but I still don't see
the relevance of this.

 > I stated this conjecture only because you demanded one,
 > but I would really rather return to my original general statement:

OK, let's drop the discussion of your conjecture, which makes no sense
to me anyway.

 > 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.

Why?  I don't see that such a relationship would have a bearing on any
important f.o.m. issue.  It seems *much* more important to find good
examples of finite combinatorial statements that require large
cardinals to prove them.

 > I have always been puzzled as to why you considered the particular
 > result of Harvey such a key result in the completeness program.

The reason I view Harvey's independence result as key is that it is
the state of the art vis a vis the program that I mentioned, i.e. to
extend the incompleteness phenomenon into finite combinatorics, or
more specifically, to find finite combinatorial statements which are
independent of ZFC, or ZFC plus large cardinals.  By state of the art
I mean the best result that is known at the present time.

 > 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.

I'm looking forward to it.  I wonder how you are going to express your
thoughts on these matters without using any informal concepts.

-- Steve

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