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From: Michael Shulman <shulman@uchicago.edu>
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Subject: Re: "Kantor dust"
Date: Wed, 11 Feb 2009 10:11:44 -0600
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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