[FOM] PA Incompleteness

[Arnon Avron]

On Fri, Oct 19, 2007 at 02:45:18PM -0400, Harvey Friedman wrote:

> >> 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. 

True. Still, it seems to me that the expression
"occurring along the path of mathematics
according to such systematic motivations and modus operandi"
demands more clarification before one tries to provide
an answer to the above question. Thus it seems that you think 
that statements like the consistency of PA are not considered now by 
"normal mathematicians" as occurring along that path, even though 
historically they definitely occur on it at the time Hilbert and 
his school pursued Hilbert Program, and even though other propositions
about consistency of systems were frequently in the past
tackled and solved by "normal" mathematicians (for example: the 
consistency of non-Euclidean geometries. Another example is provided
by the celebrated results of Goedel and Cohen, which were after all
about the consistency of ZF+GCH and ZF+\neg GCH, respectively).

> > In any case,
> > I do not see how can anyone justify a negative answer to this question

> > ...
 
> A huge proportion of questions across many disciplines have this character.
> That it looks hopeless to prove one side, but not hopeless to prove the
> other. This is totally standard. 

True again. I did not mean to use this observation 
as an argument why the question is ill-defined.
My point was that with all respect, when "normal"
mathematicians state with confidence  that the answer to the above
question is negative, they are not behaving as mathematicians or
even scientists (whether normal or abnormal).

> > 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.

Do you really think so? I would like to hear your reasons (this is 
*not* meant as a polemic. I personally give at present 50% chances to
the possibility that Goldbach's conjecture is a true proposition 
which is underivable in PA. However, I am open to good arguments that 
might change the  probability I now assign to this possibility). 

> > 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.

In principle - you are right. However, since the belief in a negative
answer is not based on rational arguments,
I am not sure that what should in principle be sufficient will really 
suffice.  (BTW, what about the alleged proof of FLT? Does anybody try to
determine at last in what system is it valid?)

> > 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? 
 
> 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.

I would not call it "wisdom". It is nothing more than wishful
thinking, since no rational argument for this "wisdom" has ever
been given. 

 Now the main importance of the incompleteness phenomena
is the "no-solution" prospect it *always* does provide in each 
individual case. Whatever "normal" mathematicians say, 
deep inside they now *know* (since they are 
mathematicians!) that without a convincing proof
to the contrary, they usually cannot be sure that the problem
they are trying to solve is indeed decidable within the
framework in which they work. But what can they do
about it, except for hoping that this is not the case,
and  acting as if they are convinced that it is not?

So I repeat my question, to which you gave no answer: 
In what way do you expect the incompleteness phenomena
to affect the behavior or research of "normal" mathematicians?

One remark: one way in which it has already affect 
"normal" mathematics is by throwing from it propositions
that have been proved to be undecidable. Are there now
(what you call) "normal" mathematicians that are trying
to solve the problem of CH? Will anybody put it now
anywhere on the list of the most important open problems
of mathematics? Yet CH certainly was a problem of
"normal" mathematics at the time Hilbert formulated his
famous list of problems!

This bring me back to my first reservation: the term 
"normal mathematics" is *too* ill-defined. Please give me 
some *objective* criteria for it.

Arnon Avron