FOM: Incompleteness program

[Joseph Shoenfield]

      This is a reply to Harvey's reply of 14 Sep.
      About Godel: I think there is no doubt that Godel was very
interested in the consequences of large cardinal axioms.    As to
what kind of consequences, he says the should be "far outside the
domain of [large cardinals]" and "where the meaning and ambiguity
of the concepts entering into them can hardly be doubted".   Of
course, he did not intend these quotes to delineate the conse-
uences exactly; but I think that any consequence expressible in the
language of PA would satisfy his conditions.   Because of this, I
do not think the quotations support the incompleteness program
which you and Steve favor.
     I now understand the reasons for your remarks about quotes B
and C; but I do not think these reasons imply that B and C are
either false or misleading.   I do not think people take "propo-
sition about Diophantine equations" to imply that the equations
must be "reasonable", nor do I think that restrict such propositions
to be statements that a particular Diophantine equation has a
solution.
     What I meant by "better" in my remarks on Paris-Harrington is
not precise, much less formalizable.   In this particular case, it
means that P-H is a simple generaliztion of a well-known theorem
provable in PA.
     I don't think we have any disagreement about CH.   I thought
your previous posting implied that all the interesting consquences
of PA were in abstract set theory; but apparently you did not
intend this.
     I am sorry that we are still not communicating about regularity
conditions.   It would certainly help if you would answer a question
which I have posed in two previous postings.  You want to find
unprovable sentences satifying strong regularity conditions, so a
regularity condition must be a condition on sentences.   My question
is: if X is one of the sets of functions in your long list, what
is the condition on sentences which corresponds to X?
   You say:
     >A reader is less likely to go astray and move in less interesting
directions if they examine not just the proved theorem in isolation,
but also what the author said about what the point of the result is.
     Perhaps.   But my point was that the author is not always the
best judge of the point of the result.   I stated my Absoluteness
Theorem as a connection between constructible sets and the analytical
hierarchy; I did not realize that its impotant applications would
be to matters of absoluteness.   Moreover, one must realize that the
reader may have different interests from the author.