FOM: Incompleteness program
Joseph Shoenfield
jrs at math.duke.edu
Tue Sep 8 10:57:59 EDT 1998
I read Harvey's report on incompleteness with interest, and would
like to pose some questions to him which occurred to me. I hope he will
regard them as true questions and not as attempts to poke holes in his
program.
1. He says that the first incompleteness theorem does not provide
an intelligible example of an undecidable sentence, but the second
incompleteness theorem does. Like all logicians, I see a significant
difference in the two undecidable statements; but I wonder if
"intellgibility" is the concept best used to describe the difference.
One can (and usually does) explain the meaning of the first undecidable
sentence as: it says that it is unprovable. There are, of course,
two difficuties here: the self-referential nature of the explanation,
and the fact that the sentence can be seen to have this meaning only
by a rather lengthy analysis. Is this the sort of thing you had
in mind?
It would certainly be interesting to know why Godel bothered with
the second incompleteness theorem, which required a lot of fairly
unintersting details. (As you know, he never wrote these details up;
Bernays was the first to do so.) I always thought it was not because
of any "deficiency" in the first undecidable statement, but was due to
the significance of the second for the Hilbert program.
As a somewhat related question, could you supply some references for
your statement that Godel in his writings expressed interest in the
program?
2. Two question on what you consider to be the extent of the program.
In this report, the emphasis is almost entirely on unprovability in ZFC.
Of course, I realize the special importance of ZFC as the recognized
axiom system for mathematics. But do you regard results like the
Paris-Kirby theorem as part of the program?
Second, what about results on what might be called the degree of
unprovability? To make this more specific: if one proves an undecida-
result, is it an important part of the program to invesigate which known
independence results it is equivalent to over the base theory. I know
you have often expressed interest in such questions; I wonder if you
consider them as a basic part of the program or a sort of desirable
enhancement.
3. You say that CH was the first example of a demonstratably
independence of a mathematical result from ZFC. What about the
existence of an inaccessible cardinal? Although I can't supply
a reference, I feel von Neumann, with his good understanding of ranks
in set theory, was aware of this independence.
4. Can you explain more clearly what you mean by a regularity
condition? For example, what is the regularity condition on sentences
corresponding to one of the classes of functions which you list? In
particular, can the collection of such sentences be described by simple
syntactical conditions? Is being a finite combinatorial sentence a
regularity condition? Are there any useful criteria for recognizing
a regularity condition?
Contrary to what some of my critics appear to be saying, I am not
opposed to the use of informal concepts in fom, even when we cannot
formally analyse them. However, if such a concept is to be used as
more than a general descriptive phrase, it shold be subjected to an
informal analysis. The object of this analysis should be to insure
that all of one's reasonably receptive readers feel they understand the
meaning of the concept as used by the author. I don't think you
have given us this kind of analysis of "regularity condition".
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