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NOTE: This talk was prepared at the requrest of the
organizers of the Gödel Centenary, in case Professor Georg
Kreisel was unable to deliver his talk. Professor Kreisel
gave his talk as scheduled, and this talk was not
devliered. Excerpts were presented at our regularly
scheduled later in the meeting. REMARKS ON GÖDEL PHENOMENA AND THE
FIELD OF REALS
by
Harvey M. Friedman
Ohio State University
Gödel Centenary
April 27, 2006
I wish to make some remarks on the Gödel phenomena
generally, and on the Gödel phenomena within the field of
real numbers.
A lot of the well known impact of the Gödel phenomena is in
the form of painful messages telling us that certain major
mathematical programs cannot be completed as intended.
This aspect of Gödel – 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?” 2 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 Gödel. 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.
Consider Diophantine equations. A decision procedure for
Diophantine equations over Z or Q has been one of the holy
grails of mathematics. We know that this is impossible for
Z and suspect it is impossible for Q.
Already this bad news represents a rather substantial body
of work by many people over many years, far more than what
it took for Gödel to show this for some class of almost
Diophantine equations over Z.
The number of variables needed presently for this is 9. For
9, the degrees needed are also gigantic.
An absolutely fantastic improvement would be, say, that the
Diophantine equations over Z with 5 variables of degree at
most 10 is recursively unsolvable. 3
Want to get very dramatic? Cubics in three variables.
Such things, assuming they are true, will take massively
more effort than has been devoted thus far.
Specifically, nobody thinks that the present undecidability
proof for Diophantine equations over Z is even remotely the
“right” proof. Yet there has not been a serious change in
this proof since 1970, when it appeared.
Yet, of the so called “leading logicians” in the world, how
many have made a sustained effort to find a better proof?
How many of the leading recursion theorists, set
theorists, proof theorists, foundationalists, and, yes,
model theorists? Almost none.
We now turn to the incompleteness phenomena. The fact that
the plane flies at all comes from the original Gödel first
incompleteness theorem. That you can fly somewhere comes
from the Gödel second incompleteness theorem and
Gödel/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 60’s: 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 P01 sentence whose proof requires
far more than ZFC. Recently, I have shifted to searching
for mathematically dramatic finite sets of P01 sentences all
of which can be settled only by going well beyond the usual
axioms of ZFC. 4
In the detailed work, perfection remains elusive. So far,
the P01 (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
P01 sentences clearly have substantially less undesirable
detail.
I strongly believe in this: 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 sstatements 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 wasn’t worked on before you showed it independent?
In my own feeble efforts on Gödel 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. Let’s 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? 5
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 Gödel phenomena.
Of course, 1M years time is also absolutely nothing. This
Sun has several billion good years left. Mathematics in 1B
years time?? I’m speechless.
I now come to the field of real numbers. The well known
decision procedure of Tarski is often quoted as a deeply
appreciated safe refuge from the Gödel phenomena.
However, a very interesting and modern close look reveals
the Gödel phenomena in force.
It is known that the theory of the reals is
nondeterministic exponential time hard, and exponential
space easy. I have not heard that the gap has been
eliminated.
This lower bound is proved in a Gödelian way, drenched with
Turing machines and interpretations and arithmetizations.
Furthermore, there is another aspect that is also very
Gödelian: lengths of proofs. I think that the least length
or size of the proof/ refutation of any sentence in the
field of reals has a double exponential upper and lower
bound.
We can go further. Given a sentence in the field of reals,
what can we say about the least length/size of a
proof/refutation in ZFC? This has an exponential lower
bound.
In fact, the situation even supports the kind of Gödelian
project underway for the usual systems of f.o.m.
For instance, we can ask for a short sentence in the field
of reals all of whose proofs in ZFC with abbreviation power
are ridiculously long. We can even more ambitiously ask
that the sentence be of clear mathematical interest.
The computational complexity of the field of reals is just
barely high enough to support such results. 6
However, the “tameness” of the field of reals is most
commonly applied not to sentences in primitive notation,
but rather to sentences with mathematically convenient
abbreviations.
For example, we can add quantifiers over all polynomials in
n variables of degree £ d, for every fixed n,d. Or we can
add quantifiers over all semialgebraic functions in n
dimensions, made up of £ r algebraic components of degree £
d, with n,r,d fixed.
Presumably the resulting complexity will be far higher,
involving a substantial increase in the height of the
exponential stack. ...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
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