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/
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Received on Wed Jun 10 2009 - 18:10:14 PDT