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).