FOM: Question grown from Friedman talks

[Harvey Friedman]

Martin Davis Sat, 03 Feb 2001 13:39:12 writes:

Friedman:

>>With all due respect, what is the evidence that Godel disagreed with me on
>>this? We are talking about the actual real world real life hard nosed
>>materially existing normal mathematical literature in our libraries.

Davis:

> From G\"odel's Gibbs address:
>
>"Now if one attacks [the] problem [of axiomatizing set theory], the result
>is quite different from what one would have expected. ... one is faced with
>an infinite series of axioms,
>which can be extended further and further, without any end being visible
>... there can never be an end ... because the very formulation of the
>axioms up to a certain stage gives rise to the next axiom. It is true that
>in the mathematics of today the higher levels of this hierarchy are
>practically never used. ... it is not altogether unlikely that this
>character of present-day mathematics may have something to do with ... its
>inability to prove certain fundamental theorems, such as, for example,
>Riemann's hypothesis ..."
>
>He goes on to suggest that a "set-theoretic number theory" still to be
>developed will go much further than analytic number theory has managed to
>accomplish.

Here Godel does suggest that using higher levels of the set theoretic
hierarchy may be useful in proving "certain fundamental theorems".

However, Godel was at least as aware as we all are that almost all
fundamental theorems (conjectures) have yet to be even stated. Whereas it
is true that the only example of a fundamental "theorem" is one that
happens to be in the present literature, I believe that Godel had future as
well as present conjectures in mind.

Also, you can interpret his statements as referring to the possibility of
effectively using higher set theory - but not in the sense of it being
required. Just in the sense of its being useful. This weaker phenomenon has
already occurred - or at least nearly occurred - in connection with Laver's
work on left distributive algebras. (There may be a quibble about just what
was in the previous actual literature or not, but in any case, things
should have been).

In addition, one can interpret "higher levels of this hierarchy" in some
different ways. E.g., we all know that even the first few iterations of the
power set beyond omega are "practically anever used" and Godel could be
primarily suggesting just those for something like the Riemann hypothesis.

Of course, the reality is that there are not too many longstanding
conjectures that make for fundamental theorems, and many of the most famous
ones have been solved after a while, without resorting to any such things.
I am thinking of the four color theorem, Fermat's last theorem, Kepler's
conjecture, Bieberbach's conjecture, etc. which are concrete fundamental
theorems in normal mathematics.

In any case, if you interpret Godel in the most far reaching sense that I
think you are, and focused on present day literature, then I would regard
his use of "altogether unlikely" as entirely irrational. I am confident
that if he were alive today, he would clarify this upon request, especially
in light of the tremendous success of present day mathematics with regard
to these fundamental conjectures without resorting, in any way, shape, or
form, to higher levels of set theory.

In fact, several years ago, I talked with a well known senior mathematician
who relayed a conversation he had with Godel, in which Godel reportedly
said that he did not think that set theoretic methods beyond ZFC would be
relevant to present day conjectures. I am taking steps now to see if that
mathematician will report his conversation with Godel to the FOM.

>I don't mean to suggest that G\"odel's views may not be wrong. It's just
>that I'm at a loss to understand on what basis Harvey just dismisses the
>possibility that this view may be right.

I never dismissed the possibility. I said that its negation was far more
likely, and that holding the view that you ascribe to Godel is irrational,
in the sense of "being reasonably likely".

First of all, the overwhelming majority of people in math logic and
mathematics thinks that what you are suggesting - and arguably what Godel
is suggesting - is very unlikely. So I am in a lot of company.

Secondly, nobody is even remotely suggesting any candidate. I know of one
important exception in the logic community, and I am inviting him to state
his views on the FOM. However, his suggestions are not really in the
category of "fundamental theorems such as the Riemann hypothesis".

Furthermore, to my knowledge, nobody is actually working on showing the
independence from ZFC of any concrete problem in the normal literature.

I also stand by what I said in my previous FOM posting about the before the
fact look and feel of indications that independence may be present - as in
the situation with set theoretic problems like CH.

However, I will say that if you allow natural adjustments to the
literature, then all bets are off. That hedged belief is not irrational.
But the literature taken literally - extremely unlikely.