Bruno Marchal wrote: > > 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.
OK. But ISTM that statement implies that we are relying on an intuitive notion as our conception of natural numbers, rather than a formal definition. I guess I don't understand "unsound" in this context. > > 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. Suppose, for example, that the twin primes conjecture is undecidable in PA. Are you saying that either PA+TP or PA+~TP must be unsound? And what exactly does "unsound" mean? Does it have a formal definition or does it just mean "violating our intuition about numbers?" Brent > > 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 -~----------~----~----~----~------~----~------~--~---
