FOM: Godel quotes; regularity conditions
Joseph Shoenfield
jrs at math.duke.edu
Mon Sep 14 11:20:07 EDT 1998
Harvey's posting of 8 Sep under this title answered many of my
questions on his incompleteteness program. Here I would like to press
him on some points which I think remain unaswered, and to make a few
remarks.
I think it is clear that in his writings, Godel did not indicate
that this program was important. I do not think this indicates that
he did not think it was. Godel was very careful in his writing; he
would not publish an evaluation of the importance of a problem unless
he had something substantial to say about it.
You comment about the Godel quotes:
>B and C indicate a reluctance to make the distinctions that
are now commonly made, under which B and C would be regarded as
either false or misleading.
I would very much like to have this statement amplified and
explained.
I think everone agrees that Paris-Harrington is better that
Paris-Kirby; the improvement is Harrington's main contrbution.
However, I think it a shame that Kirby's role in this seminal
theorem seems to be ignored.
I think your two comments on the difference between CH and
inaccessible cardinals are quite pertinent. I was surprised that
in your review of your program, you do not put more emphasis on
the applicability of CH in fields which are (logically) more ele-
mentary that abstract set theory. I don't think such applications
have disappeared from mathematics; I still see papers which give
proofs which depend essentially on CH. As to the point that
inaccessibles are not provably consistent, I think that this dis-
tinction may be quite important. You may recall that some postings
by Steel concerning the possililty of establishing the truth of some
statements not provable in ZFC made a crucial distinction between
statements whose truth-value can be changed by forcing and statements
for which this was not possible.
Let me insert here a diversion concerning large cardinals. It
has been dogma in set theory since the mid-sixties that there is a
borderline in the large cardinal hierarchy, and that cardinals below
this borderderline do not contribute to mathematical (as opposed to
metamathematical) results. The line lies above Mahlo cardinals and
below Ramsey cardinals. Do you have any results or conjectures which
might change this dogma into a mathematical theorem?
I don't think I made my problem with regularity conditions clear
to you. You say that you want to find unprovable sentences satis-
fying strong regularity conditions. What is the regularity condition
on SENTENCES corresponding to one of the classes of FUNCTIONS which
you mention? I also asked if being finite combinatorial is a regu-
larity condition in your sense; it does not seem to correspond to any
particular class of functions.
The problem I suggested (finding connections between the position
of an undecidable sentence in the hierarchies and the number and kind
of large cardinals needed to prove it) and the possible solution I
suggested had, as far as I was concerned, nothing to do with your
general program. They were related to a particular result of yours.
If you examine the conjecture (with superscripts 1 changed to 0), you
will find that the case n=1 says: there is a pi-0-2 sentence undeci-
dable in ZFC which can be proved from the existence of 1 large cardinal
of a certain kind. Does this sound familiar? I realize this leaves
out what you probably consider the most important feature of your
result. But what I wanted to show was that two different people,
in analyzing a proved theorem and deciding what extensions of it are
worth investigating, may come to different conclusions.
As to the topics considered in the rest of your positing (to what
extent is fom part of mathematics; how well results in logic are
regarded by mathematicians; the future of foundations in general
and foundations of physics), I think we have each expresses our
views and little could be gained by continuing. I think that
clarifying and explaining one's position is the main object of disputes
on fom; if one succeeds in convincing his opponent to modify his views
at least a little, this is an unexpected bonus.
You ask how I found out I was wrong about the interest of reverse
mathematics to logicians outside the field. The results which
convinced me are clearly explained in your 9 Sep posting. In response
to your request for feedback: I found this posting as useful and
informative as anything you have sent to fom.
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