From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: a flaw in Cantor's Diagonal Method (cleared up) Date: 20 Jul 1999 19:49:15 -0400 Newsgroups: sci.math Keywords: real line uniquely determined by order type In article <3794AE9F.174C3D0F@ashland.baysat.net>, Nathan the Great wrote: >deja wrote: > >[snipped - standard version of Cantor's Diagonal Method] The rectangular array, infinite in two directions, with changes made to diagonal entries, is not the "standard version of Cantor's Diagonal Method". True, it is the most frequently quoted one, a special case re-made from the general form to be palatable for public consumption. Because of that, it is a frequent target of idle, ill-informed objections. The original version, quoted many times here, does not display any array, much less a diagonal, and is easy to miss by those who are hooked on the abovementioned rectangular array. A reference for the original version: Hrbacek & Jech: Introduction to Set Theory, Marcel Dekker 1999, ISBN 0-8247-7915-0 Chapter 5, Theorem 1.8. >deja, not only are you detracting from this discussion but >you're posting in a format that makes responding very >difficult. Please don't post unless you have something new >to add. If you have a proof that encompasses every >conceivable model of R please post it. If you don't, SHUT >UP! Project: Find out, in a book written by Walter Rudin, the proof that all models of the axiomatic system for a complete liearly ordered field are isomorphic (all operations, constants and relations), and realize what isomorphism is, in particualr, that it preserves cardinality. To rub it in: if one model is uncountable then so are all other models. Actually, much less is needed, as outlined in Hrbacek-Jech's book: the order type alone makes the reals unique up to order isomorphism and uncountable (the order is linear, with a countable dense subset, complete, and without endpoints). See Chapter 5, Theorems 5.7, 6.1 and the supporting Theorem 4.9, which is most instructive. And if it is of any interest: this was proved without the AC. Bored, ZVK(Slavek).