Cantor Confusion
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From: William Hughes on 13 Dec 2006 07:18 mueckenh(a)rz.fh-augsburg.de wrote: > > Unbounded is potentially infinite but it is not necessarily actually > infinite. > Please give an example of a set you consider to be actually infinite. - William Hughes
From: William Hughes on 13 Dec 2006 07:25 Han de Bruijn wrote: > William Hughes wrote: > > > Tony Orlow wrote: > > > >>Well, the proof is simple. Any finite number of subdivisions of any > >>finite interval will only identify a finite number of real midpoints in > >>that interval, between any two of which will remain more real midpoints. > >>Therefore, there are more than any finite number of real points in the > >>interval. > > > > This just shows that the number of real points is unbounded. > > It does not show it is infinite (unless of course you use the > > fact that any unbounded set of natural numbers is infinite). > > Isn't unbounded the same as infinite, i.e. = not finite = unlimited = > without a limit? > Keep your cranks straight. To TO, the set of finite naturals is unbounded but not infinite. To WM, the set of finite naturals does not actually exist. It is a potentially infinite set. To you, the set of finite integers is bounded by a largest integer so there is no unbounded set of naturals. Any statement made about an unbounded set of naturals is vacuously true. - William Hughes
From: William Hughes on 13 Dec 2006 07:30 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > A_1 = {1} > > > > A_2 = {1,2} > > > > A_3 = {1,2,3} > > > > > > > > B = {1,2,3} > > > > > > > > then B is contained in the last A_i. If there is no last A_I, then > > > > there is > > > > no A_i that contains B > > > > > > That has nothing to do with "last". > > > > If A_i contains B, then A_i contains any A_j. > > Therefore A_i is "last". > > No comment? - William Hughes
From: Franziska Neugebauer on 13 Dec 2006 07:30 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> [...] >> >> 1. You do not present a convincing definition of "number". (Most >> >> likely you have none). >> > >> > Definitions are abbreviations like the following: >> >> [too long, too old, > > Impossible. Its from my new book to appear within few days. Do you need tax-deductible expenses? > Can future be too old? Your arrow of time points in the wrong direction. >> too German; > > it is impossible to be too German. > >> no definition at all] > > It is clear that you have not understood. > >> >> 2. You do not present a convincing definition of "numbers" and >> >> "sets" which are "not fixed" or "un-fixed". >> >> >> >> 3. You do again try to discuss issues of neuro sciences >> >> (representation of abstract entities in mind (or in the brain?)) >> >> in sci math. >> > >> > Of course, because math requires mind and brain. >> >> Mind and brain and representation of (abstract) entities therein is >> still off topic in sci.math. > > This decision is > 1) wrong > 2) not yours. Which _mathematical_ institution is doing research in the field of "representation of (abstract) entities in mind or brain"? F. N. -- xyz
From: Han de Bruijn on 13 Dec 2006 07:32
mueckenh(a)rz.fh-augsburg.de wrote: > Tony Orlow schrieb: > >>Physics used to be more continuous, but atoms and quantum effects have >>been discovered. Time and space may even be discrete. Mathematics can >>reflect that, or treat things as continuous. I don't think we've >>determined for sure that nothing is continuous. Do you? > > What *in principle* can't be measured, is not existing. Sigh! The common confusion about what continuity means. Okay, I'll never become tired of repeating this over and over again: CONTINUITY IS IN THE EYE OF THE BEHOLDER. The _same_ phenomenon in nature may be discrete as well as continuous. Fluid Flow of water consists of a zillion molecules, hence it is Discrete. But it is described with the continuum equations of Fuid Dynamics, hence it is Continuous. A less well-known example is the Fluid Tube Continuum model, by which I've been able to calculate the flow and temperature distributions in a shell and tube heat exchanger. It's found at: http://hdebruijn.soo.dto.tudelft.nl/www/programs/pascal.htm#Nerat Graphically: http://hdebruijn.soo.dto.tudelft.nl/www/programs/plaatjes/slide01.htm It is even possible to calculate critical values for certain variables in such models. Beyond such values the continuum model breaks down and the discrete substrate becomes sensible. See: http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft http://hdebruijn.soo.dto.tudelft.nl/jaar2004/IHXTAK.pdf Let's hope that my drops of water can ultimately wear away the hardest stone. Han de Bruijn |