[FOM] PA Incompleteness

Arnon Avron aa at tau.ac.il
Fri Oct 19 07:43:33 EDT 2007

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 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).

 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. 
> 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.

  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). 

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