On Tue, Feb 10, 2009 at 4:18 PM, Prof. Peter Johnstone <P.T.Johnstone@dpmms.cam.ac.uk> wrote:
Many thanks for an interesting thread! Just out of curiousity, is the Conway construction clearly identified with the Dedekind reals? How does the construction fit into the constructivist debate?
The trouble with the Conway construction is not that it's non- constructive, but that it isn't (in any reasonable sense) a construction of the reals. If you stop it at the point when it finally constructs the real numbers 1/3, \sqrt{2}, \pi and so on, then it has also succeeded in constructing lots of non-real numbers like \omega, 1/\omega, 1/2-1/\omega and so on. So how do you distinguish the numbers you want from the ones you don't?
And it isn't just "infinite" and "infinitesimal" numbers like \omega and 1/\omega that come along for the ride, either. Classically, the copy of the natural numbers sitting inside the surreal numbers is actually the *finite ordinal numbers*, and constructively there are many more ordinal numbers below \omega than there are natural numbers. For instance, { 0 | P } where P is an undecidable statement, is a perfectly good ordinal number that lies "somewhere between 0 and 1." Mike