From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2150 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Cauchy completeness of Cauchy reals Date: Wed, 5 Feb 2003 12:56:33 -0800 Message-ID: <20030205205632.GC26302@math-rs-n01.ucr.edu> References: <3E36ED4F.4070807@kestrel> <1043829334.3e37925619e77@mail.inf.ed.ac.uk> <3E3F849F.5080704@kestrel.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018450 2737 80.91.229.2 (29 Apr 2009 15:20:50 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:20:50 +0000 (UTC) To: CATEGORIES mailing list Original-X-From: rrosebru@mta.ca Thu Feb 6 16:43:49 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 06 Feb 2003 16:43:49 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18gsnO-0002dk-00 for categories-list@mta.ca; Thu, 06 Feb 2003 16:38:54 -0400 Content-Disposition: inline In-Reply-To: <3E3F849F.5080704@kestrel.edu> User-Agent: Mutt/1.4i Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:2150 Archived-At: Dusko Pavlovic wrote: >Alex Simpson wrote: >>Somebody wrote: >>>that 1/2 + 1/4 +...+ 1/2^k > 1-e. in other words, that there is k >>>s.t. 1/2^k < e. this is *equivalent* to markov's principle. >>The property quoted is in fact a trivial consequence of intuitionistic >>arithmetic alone. It has nothing to do with Markov's principle. >for a real number e, i am pretty sure that the above is equivalent with >markov's principle. it must be in books, but i think you can't miss the >proof if you try. I don't remember the original context, so I don't know who's right, but the answer depends on what sort of real number e could be. It can't be 0, for example, so what can it be? * If e > 0, then work with 1/e and use the Archimedean property; Bishop would recognise and accept this proof. * But if you only know that e <= 0 is false, then you need Markov's principle to deduce e > 0. -- Toby