From archibal@fresco.Math.McGill.CA Mon Aug 23 00:19:19 2004 Received: with ECARTIS (v1.0.0; list lojban-beginners); Mon, 23 Aug 2004 00:19:19 -0700 (PDT) Received: from fresco.math.mcgill.ca ([132.206.150.41]) by chain.digitalkingdom.org with esmtp (Exim 4.34) id 1Bz96t-0007jk-0T for lojban-beginners@chain.digitalkingdom.org; Mon, 23 Aug 2004 00:19:19 -0700 Received: (from archibal@localhost) by fresco.Math.McGill.CA (8.11.6/8.11.6) id i7N7Int25428 for lojban-beginners@chain.digitalkingdom.org; Mon, 23 Aug 2004 03:18:49 -0400 Date: Mon, 23 Aug 2004 03:18:49 -0400 From: Andrew Archibald To: lojban-beginners@chain.digitalkingdom.org Subject: [lojban-beginners] Re: logical proofs? Message-ID: <20040823031849.A32026@fresco.Math.McGill.CA> References: <1093232310.412966b640996@www.gulik.co.nz> <20040823041958.GD3257@chain.digitalkingdom.org> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.2.5.1i In-Reply-To: <20040823041958.GD3257@chain.digitalkingdom.org>; from rlpowell@digitalkingdom.org on Sun, Aug 22, 2004 at 09:19:58PM -0700 X-archive-position: 718 X-Approved-By: archibal@math.mcgill.ca X-ecartis-version: Ecartis v1.0.0 Sender: lojban-beginners-bounce@chain.digitalkingdom.org Errors-to: lojban-beginners-bounce@chain.digitalkingdom.org X-original-sender: archibal@math.mcgill.ca Precedence: bulk Reply-to: lojban-beginners@chain.digitalkingdom.org X-list: lojban-beginners On Sun, Aug 22, 2004 at 09:19:58PM -0700, Robin Lee Powell wrote: > On Mon, Aug 23, 2004 at 03:38:30PM +1200, mikevdg@gulik.co.nz wrote: > > Is it possible to do logical proofs in lojban? > > Absolutely. Give me a short one, and I'll translate it (*much* too > tired to do this now). Proof that there are infinitely many prime numbers: Suppose that we have a finite list of prime numbers. Then construct the number that is one greater than the product of all the prime numbers in the list. This number is, by a previous result (you could prove it if you wanted), guaranteed to be divisible by some prime number - perhaps itself. But the number is not divisible by any of the numbers in our finite list of primes, so there must be at least one prime not on the list (although it may not be the new number). So no finite list can contain all the prime numbers. If you'd like this a little more self-contained (you'll still need to depend on some properties of multiplication, addition, and ordering of the natural numbers) you can prepend the definition of a prime number: a natural number (i.e., positive integer) such that it can be written as a product of natural numbers in exactly one way, as one times itself. It then follows by induction that every number is divisible by a prime number. It might be interesting to write this as a proof by contradiction. Such proofs are always potentially confusing, with their temporary assumption of some "fact" which turns out to be impossible. Andrew Archibald