On 7/14/09, Dan Asimov <dasimov@earthlink.net> wrote:
I am assured by logicians that in any axiomatic system rich enough to include number theory, there are propositions that can be neither proved nor disproved, regardless of whether they are "really" true or "really" false. ... BUT -- what if a hypothetical infinite check of all possible twin primes shows that TWP is true . . . yet we choose to add as an additional axiom its negation ~TWP. Then there is a model of this system, despite the fact that it is WRONG.
Would that system really exist?
Sure the system (qua theory) exists: you have just created it. It just wouldn't (eventually) prove to be of much use! [Perhaps instead a more resonant proposition to consider in this connection might be the Riemann conjecture. If some heroic computation eventually discovers a zero away from the critical line, then an awful lot of number-theoretic papers will go down the tube --- tough!] In clarifying my own stance on such questions, I consider it important to keep in mind that retreating into pure mathematices does not enable us to escape engineering. All that changes is that both theory and application are abstract --- here Peano arithmetic applied to |Z, as opposed to (say) structural mechanics applied to a wobbling bridge. WFL