FOM: Crucial f.o.m. issue

[Harvey Friedman]

This is a reply to Shoenfield 3:27PM 9/8/98. Before starting, let me say
that J. Shoenfield has proved (1961) an absolutely crucial result, called
Shoenfield absoluteness, which is an absolutely crucial tool in the
subsequent development of several aspects of f.o.m. The result is
mathematically beautiful and deep, and has also been used in a virtually
infinite variety of situations in the development of set theory.

>I regret that in my possible conjecture, all of the 1's should have
been 0's. By a strictly pi-0-n+1 sentence I mean one which is not
>pi-0-n or sigma-0-n.

OK.

>I stated this conjecture only because you
>demanded one, but I would really rather return to my original general
>statement: one should look for a result which relates the position of an
>undecidable statement in the arithmetical or analytic hierarchy and the
>number and kind of large cardinals needed to prove it.

This is an **idiosyncratic** and wholly **irrelevant** technical side issue
that is entirely unrelated to any legitimate issue I know of in f.o.m.
Whereas mathematics virtually runs on pi-0-1, pi-0-2, maybe pi-0-3
theorems, there aren't any pi-0-148 sentences or higher in the entire
standard mathematical literature.

The crucial issue for f.o.m. and the future of set theory is, generally,
this: does the incompleteness phenomena touch purely arithemetic
mathematical sentences? And purely arithmetic mathematical sentences of the
logical structure that are encountered in mathematics? [These are pi-0-2 or
pi-0-1; sometimes pi-0-3 sentences. And when a pi-0-3 sentence comes up,
there is great interest in giving a pi-0-2 form (e.g., Falting's theorem).
These reflect a crucial aspect of the real structure of actual
mathematics.] And in what way does it touch? Can new axioms going beyond
ZFC be used in an essential way in such domains? And in what way?

This issue has been entirely crucial for f.o.m. since the 1930's and
especially since the 1960's, after it became clear how incompleteness
affects set theoretic statements, and the essential powerlessness of
forcing in connection with absolute statements.

The crucial difference between, on the one hand, purely arithmetic
mathematical sentences, and sentences like CH, or ones about projective
sets, is completely obvious - and must be to you as well. That's way I find
your comments on this obvious and well known program going back several
months here on the FOM **entirely incomprehensible.**

What actually occurs in the program (for some of the results) is something
much more interesting and relevant than the technical phenomenon that you
relate in the second quote above:

1. The pi-0-2 and pi-0-1 statements have a universal numerical quantifier
in front which ranges over dimensions.
2. As the dimension goes to infinity, higher and higher large cardinals are
needed in order to prove the statement in any reasonable length.
3. There are natural variants so that when the dimension is fixed, the
statement remains pi-0-2 or pi-0-1 (instead of becoming sigma-0-1 or
delta-0-0). In this case, as the dimension goes to infinity, higher and
higher large cardinals are needed to prove the statement.

> I have always been puzzled as to why you considered the par-
>ticular result of Harvey such a key result in the completeness program.

[Do you mean incompleteness instead of completeness?]

When I was at Stanford in the late 60's, Cohen said: "but the real problem
(now) is to get some independence results involving the integers, and
develop techniques for handling nonstandard models for that purpose."

Obviously, the **kind** of results I am doing is completely crucial in the
strongest possible sense for f.o.m. **THE ONLY REMAINING ISSUE IS HOW
NATURAL AND SUGGESTIVE AND CONVINCING ARE THE EXAMPLES THAT I DEVELOP?**
What other issue is there about this?

That's why I spent 25 or 30 years perfecting and improving such results,
and will likely continue indefinitely.

>     I agree that it would not be useful for us to plunge into a debate
>on the relative virtues of the two programs.

It would be useful to find out what on earth you are saying. It's
completely incomprehensible to me at the most fundamental level. I am
convinced that not a single person on the FOM has the vaguest idea what you
are driving at - I certainly don't. There is probably some sort of
****massive trivial misunderstanding**** which might be easily clarified.
Perhaps like you having put 1's in instead of 0's as you agreed they should
be (see first paragraph of this posting). Please explain.