FOM: Martin-Steel theorem

Stephen G Simpson simpson at
Sat Sep 12 13:56:51 EDT 1998

Joseph Shoenfield writes:
 > When I say a sentence is pi-0-n+1 but not pi-0-n, I mean it is not
 > equivalent (semantically) to a pi-0-n sentence. 

What do you mean by "equivalent (semantically)"?  For a Platonist, any
two true sentences are equivalent in the sense that they have the same
truth value in "the real world", but that's probably not what you have
in mind here.  I genuinely don't understand what you are getting at.
In my posting of 9 Sep 1998 13:18:12, I suggested that you might be
referring to provable equivalence in ZFC, but apparently that's not it

 > I don't know why we must proceed with this ridiculous discussion,
 > except that you seem determined to show that what you call my
 > conjecture is incomprehensible.

When I said "your conjecture", I should probably have said instead
"your suggestion to extend Harvey's results in a direction analogous
to Martin-Steel".  We don't have to proceed with the discussion of
that suggestion, especially if you don't have anything specific in
mind.  I have already agreed to your proposal to drop that discussion.

 > As to whether the particular result of Harvey's which began this
 > discussion is a key result in his program, I find it amazing that
 > you would give as a reason that it is the best result known at the
 > present time. 

Why are you amazed?  I see Harvey's result on trees as a key part of
the mathematical incompleteness program at the present time, because
it was motivated by the program and is the best known result in the
program at the present time.  That being the case, it seems reasonable
to evaluate the specific result in tandem with an evaluation of the
program of which it is a part.  I see no point in artificially
separating the two at the present time.

 > This suggest that as soon as Harvey proves a better result, this
 > result will cease to be a key result.

Well, that could conceivably happen.  But I think you yourself implied
that it would be "a waste of time" to try to evaluate a research
program in the absence of specific results.

 > You also seem to suggest that Harvey's program in only concerned
 > with finite combinatorial statements; but in his recent exposition
 > of his program, such statements enter only as a particular example
 > of what he want to investigate.

I think we are in agreement on this.  I said explicitly in 5 Sep 1998
00:20:43 that the search for finite combinatorial independence results
is only part of the general program of finding mathematical
independence results in various branches of mathematics.

 > As I have remarked at least twice in recent postings, I have no
 > objection to informal concepts, except in some very special
 > circumstances which I have described in great detail.

Good, I'm glad.  But you objected strenuously to my use of well
understood informal concepts such as "understandable", "finite
combinatorial", etc., in formulating the interest of Harvey's results.
Therefore, I am wondering what kind of informal concepts, if any, you
will allow yourself to use in formulating the interest of

 > besides indicating that you have not read my postings very
 > carefully,

I think I've read your postings with reasonable care.  E-mail is not a
perfect medium of communication, so we all ought to exercise patience.

 > might also lead me to wonder if the previous statement
 > was not (like the reply of the editors in a hypothetical happening
 > which we discussed) merely politeness.

I am genuinely eager to hear and analyze your explanation of why
Martin-Steel is a good theorem, even if the explanation uses informal

-- Steve

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