On 10 Jun 2009, at 02:20, Brent Meeker wrote:
> I think Godel's imcompleteness theorem already implies that there must > be non-unique extensions, (e.g. maybe you can add an axiom either that > there are infinitely many pairs of primes differing by two or the > negative of that). That would seem to be a reductio against the > existence of a hypercomputer that could decide these propositions by > inspection. Not at all. Gödel's theorem implies that there must be non-unique *consistent* extensions. But there is only one sound extension. The unsound consistent extensions, somehow, does no more talk about natural numbers. Typical example: take the proposition that PA is inconsistant. By Gödel's second incompletenss theorem, we have that PA+"PA is inconsistent" is a consistent extension of PA. But it is not a sound one. It affirms the existence of a number which is a Gödel number of a proof of 0=1. But such a number is not a usual number at all. An oracle for the whole arithmetical truth is well defined in set theory, even if it is a non effective object. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
