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 tholol.

Posted by Jeff
on 20050724 15:23:57 