From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3308 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Construction of a real closure Date: Sat, 6 May 2006 09:53:49 -0400 (EDT) Message-ID: <19360.2776191712$1241019220@news.gmane.org> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019220 8184 80.91.229.2 (29 Apr 2009 15:33:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:33:40 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Sat May 6 11:49:07 2006 -0300 X-Keywords: X-UID: 228 Original-Lines: 17 Xref: news.gmane.org gmane.science.mathematics.categories:3308 Archived-At: John's post was interesting, but a couple of things bother me. Since there are explicit enumerations of Q[x] and you can order the roots of any polynomial by their order and adjoin the roots in order, how can you get an uncountable closure? Another question (for Q, although any Archimedean field should work) what if you use Cauchy sequences generated by Newton iteration, having isolated the real roots using Sturm and sticking to an interval in which the derivative is positive (I am thinking of a monic polynomial that is increasing in the end)? Incidentally, AC doesn't bother me at all, but constructive methods are interesting in their own right. Michael