FOM: Incompleteness program
Joseph Shoenfield
jrs at math.duke.edu
Sun Sep 20 17:54:06 EDT 1998
This is a reply to Harvey's reply to my Sep 17 posting.
Let me first give a summary of your incompleteness program as I
have come to understand it in our discussions. The principal object
is to improve Godel's second incompleteness theorem but finding new
statements unprovable in ZFC which are, in some sense, simpler than
ConZFC. The most important criterion of simplicity is that the
setences be mathematical. This can be explained at least roughly
as: the sentences should be similar to results which core mathema-
ticians are proving or attempting to prove. If there is anything
seiously misleading or false in this description, let me know.
In the Godel quotes he discusses certain sentences unprovable in
ZFC but provable from large cardinal axioms. It is clear that these
sentences are of the form ConZFC', where ZFC' is ZFC with some large
cardinal axioms. He makes various comments intended to show that
these sentences are simpler than the large cardinal axioms; the sense
of simpler is made clear by the comments. There is nothing that
indicates that he thinks it would be valuable to find undecidable
sentences which are still simpler. Thus he clearly believes that
these sentences are in the field of Diophantine equations, and shows no
desire to replace the Diophantine equations by "reasonable" ones. He
shows no interest in searching for mathematical (in the above sense)
sentences. In fact, the following quotation from the Gibbs lecture
could indicate that the opposite is true:
>It is safe to say that 99.9% of present-day mathematics is con-
tained in the first three levels of this hierarchy. However, this
is a mere historical accident, which is of no importance for questions
of principle.
My description of better came from the Paris-Harrington article;
if you find it unsatisfactory, perhaps you should tell Leo instead of
me. I don't see why I am coming under attack for agreeing with your
unexplained statement that PH is beteer that KP; perhaps you are
in an attacking mood.
I find an answer to my question on regularity conditions in your
statement: "I want to find unprovable sentences involving only
objects ...". However the meaning of "involving" is not clear.
I can think of various ways of determining what objects are involved
in a sentence, and I think the differences might be crucial for your
program.
Let me further clarify my feelings on vague concepts. I think
it is not at all objectionable to use vague terms like "simpler" and
"involving" in the description of a program. I think it is often
desirable to postpone consideration of the precise meanings of these
concepts until the program progresses, since the results may give
clues as to what precise meaning these concepts should have.
But I think it is wise if the author of the program calls attention
to these imprecisions, lest he later be accused of failing to stick
to his stated program.
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