FOM: Reply to Steve
Joseph Shoenfield
jrs at math.duke.edu
Sat Sep 12 11:52:03 EDT 1998
This is a reply to Steve's posting of Sep 9 entitled Martin-Steel
Theorem.
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. Under the more literal
interpretation you seem to favor, no pi-0-n+1 sentence is pi-0-n; so
your comments about dummy quantifiers is irrelevant. If I adopted
your interpretation, I could say that your claim that Harvey's tree
principle is pi-0-2 is wrong, since his principle is stated in English
and English sentences are not classified in the arithmetical hierarchy.
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.
As to the problem of finding connections between the position of
an undecidable sentence in the hierarchies and the large cardinals which
suffice to prove it: My object in formulating it was to demonstrate
that in analyzing a proved theorem to find out its foundational signi-
ficance and get ideas for further research, different people can come
to quite different conclusions without either of them being wrong. I
find the question interesting; more precisely, I think that an answer
to it would be of considerable interest, provided it had a clear content
and was not contrived. I think experts in large cardinals would agree.
To try to compare its importance with that of some quite different problem
before one had the answer would be a waste of time. In general, I think
comparing the importance of theorems on quite different subjects is
useless (except as an entertaining topic at coffee hour); it depends too
much on each individual's criteria of importance.
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. This suggest that as soon as Harvey proves a better result, this
result will cease to be a key result. 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.
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. I wonder if you reailze that
the final statement in your posting, besides indicating that you have
not read my postings very carefully, 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.
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