Gödel proved that there are true sentences that cannot be proved.

Suppose I told you that the Goldbach conjecture is one of those. (The Goldbach conjecture supposes that every even integer number can be expressed as the sum of two odd primes.)

Is that logically possible? (And, no, I haven't proved it!)

The Riemann Hypothesis (RH) is another famous open problem.  Apparently some mathmeticians have assumed the truth of RH to make further  hypothesis.  (I suppose in number theory).  Suppose we admit the Goldbach Conjecture (GC) as a new axiom to construct a new system.  Suppose we construct another system with ~GC as an axiom.  (~GC says the conjecture is false).  If one system led to logical contradicitions could we draw any conclusions?  We would have to assume consistency in the original system tho-lol.

Posted by Jeff on 2005-07-24 15:23:57