On Mar 25, 2:10 pm, barry barrett <barry3...@hotmail.com> wrote:
> There are two proofs here. 1. The incompleteness of P and 2. The General Incompleteness of P-like systems. > I still don't see how you can call the proof of 2 (in Godel's original paper) "constructivist" by any > definition of "constructivist" I've see.
Whether in the original paper or not, all we have to do is define 'P- like system' (such as we do when we say such things as "a recursively axiomatized theory adequate for a certain amount of arithmetic" (where, 'a certain amount of arithmetic' may be given some definition)), then, for any arbitrary consistent P-like system, we use Godel's method to construct an undecidable sentence.
That is, GIVEN an arbitrary hypothetical consistent P-like system talked about with the variable 'S', we can not only prove that there is an undecidable sentence for S, but moreover we can DEFINE that undecidable sentence with a definition that has no free variables other than 'S'. What version of constructivism disallows that as constructive?
> Maybe there are degrees of constructivism. I can imagine a certain type of constructivist accepting this. > To put it succinctly: how could you accept this, but not accept proof by contradiction?
What does that have to do with constructivist rejection of non- constructivist proof by contradiction?
> "Providing a constructive method" to prove x (in all cases where 1 & 2 apply) is not the same as providing > a constructive proof of x for all those cases.
What does that have to do with the claim that Godel's proof (given that we provide some formulations of 'P-like system') is constructive? Given an arbitrary P-like system S, we construct an undecidable sentence for S. Of course, there are infinitely many such systems S, so we can't, as humans, perform this construction for each individual such system. But constructivism doesn't require us to that. It is sufficient that we prove for an arbitrary S and then apply "universal generalization" to conclude that for ANY such system S there is an undecidable sentence.
> So I half agree with you. It is somewhat constructive because it at least provides you with the method to > construct the undecidable proposistion for a any P-like system.
I don't know what "other half" there is.
> Would you at least grant me that the proof that 1. ((P is incomplete)) is more constructive in nature then > 2. ((the proof of general completeness))?
I don't know about DEGREES of constructivity (there very well could be such a thing though I am not informed as to it), but the general proof is constructive by ordinary constructivism such as by the BHK interpretation.