[FOM] PA Incompleteness

[Harvey Friedman]

On 10/19/07 7:43 AM, "Arnon Avron" <aa at tau.ac.il> wrote:

> On Sun, Oct 14, 2007 at 10:02:58AM -0400, Harvey Friedman wrote:
>> 
>> Why should we care about "normal" mathematicians? Mathematics is a subject
>> with some sort of systematic motivation and modus operandi - although it is
>> far less clear what this is than it ought to be, as mathematicians do not
>> seem to want to write or lecture directly about it in any serious way.
>> 
>> So a very legitimate question is whether incompleteness is actually
>> something that occurs along the path of mathematics according to such
>> systematic motivations and modus operandi.
> 
> I am not sure that this is a well-defined question.

In intellectual activity, or even in scientific activity, there are almost
no well defined questions. This question is comparatively well defined.
E.g., it is far more well defined than more usual formulations like "there
is a mathematically natural example of an independent sentence", etcetera.

> In any case,
> I do not see how can anyone justify a negative answer to this question
> except as a declaration of personal faith. A mathematician who
> claims with confidence that incompleteness will never occur
> along the path of "normal" mathematics is certainly not
> doing this on the basis of the norms according to which "normal"
> mathematicians (or even scientists in general) accept propositions,
> especially those that have concrete mathematical consequences
> (Thus it follows from such a claim
> that Goldbach's Conjecture is decidable in PA - although nobody has
> a mathematical proof of this decidability proposition).

A huge proportion of questions across many disciplines have this character.
That it looks hopeless to prove one side, but not hopeless ot prove the
other. This is totally standard. E.g., can be harness the Sun's energy to
solve the world's energy problems cheaply for the next 1M years?
> 
> Of course, the fact that nobody can justify a negative answer
> is not a justification for a positive answer. But a convincing
> justification of a positive answer is at least possible. Thus
> it would suffice for it to show that if Goldbach's conjecture is true
> then it cannot be proved in PA.

Obviously this would be sufficient. But also not likely to be true.
> 
>> Actually, I have no doubt that it does, but demonstrating this in a totally
>> convincing way is an important and fair challenge.
> 
> It seems to me that only a proof that one of the famous
> open problems of number theory is undecidable in PA can
> serve as such a totally convincing demonstration.

This is not the case. For instance, one can set up all kinds of modified
forms of famous open problems in number theory that would clearly suffice.
Also, it is easy to list hundreds if not thousands of theorems in the
literature that would suffice - if they happened to have been undecided in
PA and we could show that they were.
> 
> However, what I do not fully understand is why such a demonstration
> is an important challenge for itself. My question is:
> In what way does Harvey expect it to affect the behavior or research of
> "normal" mathematicians? (some hints from past postings
> suggest that Harvey's real goal is to convince "normal"
> mathematicians to start using strong axioms of strong
> infinity. If so, I do not share this goal. On the contrary).

It is an obviously completely fundamental issue in the foundations of
mathematics. Until it is solved, there is the prospect that the
incompleteness phenomena are not relevant to the development of mathematics
- not only now, but in the future.

Again, this is a totally standard situation. Conventional wisdom is that the
incompleteness phenomena is intrinsically irrelevant to mathematics.

Harvey Friedman