[FOM] Godel Centenary Meeting 2

[Harvey Friedman]

Continuation of http://www.cs.nyu.edu/pipermail/fom/2006-May/010485.html

I continued my talk with an excerpt from the talk I didn't give (to replace
Kreisel in case he wasn't able to speak. He did speak.)

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A lot of the well known impact of the Godel phenomena is in the form of
painful messages telling us that certain major mathematical programs cannot
be completed as intended.

This aspect of Godel ­ the delivery of bad news - is not welcomed, and
defensive measures are now in place:

1. In Decision Procedures. "We only really wanted a decision procedure in
less generality, closer to what we have worked with successfully so far. Can
you do this for various restricted decision procedures?"

2. In Decision Procedures. "We only really wanted a decision procedure in
less generality, closer to what we have worked with successfully so far.
Here are restricted decision procedures covering a significant portion of
what we are interested in."

3. In Incompleteness. "This problem you have shown is independent is too set
theoretic, and pathology is the cause of the independence. When you remove
the pathology by imposing regularity conditions, it is no longer
independent."

4. In Incompleteness. "The problem you have shown is independent has no
pathology, but was not previously worked on by mathematicians. Can you do
this for something we are working on?"

Of these, number 3 is most difficult to answer, and in fact is the one where
I have real sympathy. So I will focus on 1,2,4.

I regard these objections as totally natural and expected.

When the Wright Brothers first got a plane off the ground for long enough to
qualify as "flight", obvious natural and expected reactions are:

Can it be sustained to really go somewhere?

If it can go somewhere, can it go there in a reasonable amount of time?

If it can go there in a reasonable amount of time, can it go there safely?

If it can go there safely, can it go there economically?

The answer to these and many other crucial questions, is YES. In fact, a
bigger, more resounding YES then could have ever been imagined at that time.

But to establish yes answers, there had to be massively greater amounts of
effort by massively more people, involving massive amounts new science and
engineering, than were involved in the original breakthrough.

And so it is with much of Godel. To reap anything like the full consequences
of his great insights, it is going to take far greater efforts over many
years than we have seen.

I now turn to the incompleteness phenomena. The fact that the plane flies at
all comes from the original Godel first incompleteness theorem. That you can
fly somewhere important comes from the Gödel second incompleteness theorem
and the Godel/Cohen work. Upon reflection after many years, we now realize
that we want very considerable flexibility in where we can fly.

In fact, there will be a

virtually unending set of stronger and stronger requirements as to where we
want to go with incompleteness.

I have merely scratched the surface of non set theoretic destinations for
incompleteness, for 40 years. Almost alone ­ I started in the late 60¹s: in
1977 I was not alone (Paris/Harrington for PA).

The amount of effort devoted to unusual destinations for the incompleteness
phenomena is trivial. Well, *I* might be exhausted from working on this, but
what does that amount to compared to, say, the airline industry after the
Wright Brothers? Zero.

Most of my efforts have been towards finding that single mathematically
dramatic Pi01 sentence whose proof requires far more than ZFC. Recently, I
have shifted to searching for mathematically dramatic finite sets of Pi01
sentences all of which can be settled only by going well beyond the usual
axioms of ZFC. 

In the detailed work, perfection remains elusive. So far, the Pi01 (and
other very concrete) statements going beyond ZFC still have a bit of
undesirable detail. There is continually less and less undesirable detail.
The sets of Pi01 sentences clearly have substantially less undesirable
detail.

I strongly believe in this extremely strong major conjecture:

EVERY INTERESTING SUBSTANTIAL MATHEMATICAL THEOREM CAN BE RECAST AS ONE
AMONG A NATURAL FINITE SET OF STATEMENTS, ALL OF WHICH CAN BE DECIDED USING
WELL STUDIED EXTENSIONS OF ZFC, BUT NOT WITHIN ZFC ITSELF.

Recasting of mathematical theorems as elements of natural finite sets of
statements represents an inevitable general expansion of mathematical
activity. This applies to any standard mathematical context. This program
has been carried out, to some very limited extent, by BRT ­ details will be
presented Saturday.

Now concerning the issue of: who cares if it is independent if it wasn¹t
worked on before you showed it independent?

In my own feeble efforts on Godel phenomena, sometimes it was worked on
before. Witness Borel determinacy (Martin), Borel selection (Debs/Saint
Raymond), Kruskal's tree theorem (J.B. Kruskal), and the graph minor theorem
(Robertson/Seymour).

Mathematics as a professional activity with serious numbers of actors, is
quite new. Let¹s say 100 years old ­ although that is a stretch.

Assuming the human race thrives, what is this compared to, say, 1000 more
years? Probably a bunch of minor trivialities in comparison.

Now 1000 years is absolutely nothing. A more reasonable number is 1M years.
And what does our present mathematics look like compared to that in 1M years
time?

There is not even the slightest expectation that what we call mathematics
now would be even remotely indicative of what we call mathematics in 1M
years time. The same can be said for our present understanding of the Godel
phenomena.

Of course, 1M years time is also absolutely nothing. This Sun has several
billion good years left. Mathematics in 1B years time?? I¹m speechless.

Harvey Friedman