FOM: Question grown from Friedman talks

[Martin Davis]

At 07:18 PM 2/3/01 -0500, Harvey Friedman wrote:
>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.

Obviously, the intuition of someone who has made the kind of contributions 
that Harvey has should not be lightly dismissed.

Nevertheless, this is how things have seemed to me: We know by G\"odel that 
for any system of axioms for (say) set theory, there are Pi-0-1 sentences 
independent of those axioms. We also know that there are significant open 
questions of this form that have resisted strenuous efforts at their 
resolution. Now, a priori, this may be because the right elementary tools 
have not yet been found (as in the case of FLT for so many years), or it 
may be because one is dealing with assertions that require methods going 
beyond PA or even ZFC. How is one to decide? Does the fact that FLT 
eventually succumbed to attack by elementary methods in any way imply that 
the same will be true for, say, RH? If so, I don't see the reason.

One can imagine a completeness theorem. One would specify some criterion of 
simplicity perhaps based on complexity of expression and then prove that 
every true Pi-0-1 sentence that satisfies that criterion is provable in 
ZFC. That would shut me up - especially if RH and Goldbach's conjecture met 
the criterion. But absent such a theorem or at least some heuristic 
evidence for such a theorem, I remain agnostic.

Martin




                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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