X-Message-Number: 21979 From: Date: Sat, 14 Jun 2003 17:40:29 EDT Subject: Re: infinites --part1_129.2c324575.2c1cf04d_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit > > As it turns out, there are actually different sizes of infinity, but only > two of them. Which is the larger infinity: The number of integers, or > the number of numbers between zero and one? > > =>Cheers, > =>Matthew > I would say the second, because it has the power of the continuous. Infinites are sometime counterintuitives. Most think that infinite must be a very large number, it is not always so. Think for example about countable infinite: it is the next number after the last defined one. For example in modulo numbers, the modulo is infinite. Here is an example: The alowed number in modulo 3 are: 0 1 2. After that you get back to 0 so the endless number suite is: 0 1 2 0 1 2 0 1 2 0..... in this numbering system 3 is the infinite. At a fundamental level, the universe may be a spontaneous quantum computer unable to define number beyond 0, so here, 1 is the infinite! J. H Conway has shown that such a system can give a number field, he called ON2, so you can do any computing with this rather limited system. All numbers larger than one are Cantor's transfinites here. ON2 has some strange properties, for example the square of a quantity L is not L x L, it is: L x (L + 1), something found as an (arbitrary) law in quantum mechanics. In the set of all integers, the next one after N is N+1, because this process can be used again and again, it seems we can't get a number after the last integer. If we are at step N, it suffice to write N+0.5 to get the infinite. This fractional number can't be defined in the frame of the whole number rule. It is so one of the possible infinite of the whole numbers. What if we include in the number definition the fractional one such 2.5, the irrationals such the square root of 2, the transcendantals such pi? Then i, the square root of minus one and the basic component of complex numbers is the infinite. Well, taking into account complexes, quaternions, octonions,... what is the infinite? Octonions are 8 x 8 matrixes with non associative product: (ab)c is different from a(bc). Loops have this property with only 5 x 5 matrixes, so loops are the infinites of hypercomplexe numbering systems. The infinite has some specific properties, for example in differential geometry, if you start from a vector space E, you can define the dual space E* using the differential operators. If you do this again, you get E**. Now, if E was a flat space with a finite number of dimensions, then E**=E. On the contrairy, if E was infinite dimensional, then E** is different from E, E*** is different from E*, and so on. If E was the ordinary euclidean space it had a finite number of dimensions This is equaly true of the 4 dimensional relativistic space, but that same relativistic space put in the restricted frame of ordinary euclidean space is infinite because we can't count the dimensions up to four here. Strange it is not? Yvan Bozzonetti. "Engineers aren't boring people, we just get excited about boring things." --part1_129.2c324575.2c1cf04d_boundary Content-Type: text/html; charset="US-ASCII" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=21979