Re: Provability

In article
<rbisrael.20071023142716$7fa6@xxxxxxxxxxxxxxxx>,
Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:

Victor Porton <porton@xxxxxxxx> writes:

For famous hypothesis such as Millennium Prize problems (or to be yet
more specific for P=NP problem)...

... how we can (or cannot) be sure that the hypotheses or its negation
is provable.

We can't. In some cases we can do half of this: e.g. if the Riemann
hypothesis or Goldbach's conjecture happens to be false, then its
negation is provable. But if it is true, there is no guarantee that
there is a proof of that.

There are statements which are unprovable nor true nor false in ZFC.

For a specific example can we be sure that either P=NP or P!=NP is
provable?

No. This is even worse than the Riemann hypothesis in that regard:
thus if P=NP there is a polynomial-time algorithm for solving an NP-complete
problem, but there might not be a proof that this algorithm works.

The notion of algorithm shares an important characteristic
with the notion of theorem. A computational method is not
an algorithm until we have accepted a proof of its efficacy.

--
Michael Press
.

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