Cantor Confusion
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From: Virgil on 16 Oct 2006 20:54 mueckenh(a)rz.fh-augsburg.de wrote: > I warned you that this point is new: The edges are split in shares of > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > most people may have had the same problems as you today. I am sure you > can understand it from the written text above. (Many others have > already understood it.) Name one.
From: Alan Morgan on 16 Oct 2006 21:14 In article <1161027684.800946.299570(a)m73g2000cwd.googlegroups.com>, <mueckenh(a)rz.fh-augsburg.de> wrote: > >Alan Morgan schrieb: > >> In article <1160944143.122919.243860(a)h48g2000cwc.googlegroups.com>, >> <mueckenh(a)rz.fh-augsburg.de> wrote: >> > >> >William Hughes schrieb: >> > >> >> However, you wish to do more. You want to show >> >> that claiming "N does not have an upper bound and >> >> N exists as a complete set" leads to a contradiction.] >> >> >> >That is true too. And it is easy to see: If we define Lim [n-->oo] >> >{1,2,3,...,n} = N, then we can see it easily: >> > >> >For all n e N we have {2,4,6,...,2n} contains larger natural numbers >> >than |{2,4,6,...,2n}| = n. >> >> Agreed. > >Do you know what "for all n e N" means? There are *not any* further >natural numbers. There is no chance to increase the cardinal number. I'm not the one claiming that aleph_0 is a natural number. You are. At least, I think you are. >> >There is no larger natural number than aleph_0 = |{2,4,6,...}|. >> >Contradiction, because there are only natural numbers in {2,4,6,...}. >> >> That would be a contradiction only if Aleph0 e N, but it isn't. Your >> statement above is true for finite n. Showing that it isn't true for >> infinite n (or in the limit or whatever terminology you choose to use) >> does not produce a contradiction. > >There are no infinite n, whatever terminology you coose. And aleph_0 is >considered larger than any finite n. That is simply impossible. Impossible? That's practically the *definition* of aleph_0. You've stated something that is true for all natural numbers. It is false for aleph_0. That would be a problem if aleph_0 were a natural number. It isn't. So, no problem. >There must be finite even numbers X, larger than 2n, which complete the >set >{2,4,6,..., 2n, X }. I have no idea what you mean here. What is "complete" the set? For any fininte n there are even numbers > 2n. What is your point? Alan -- Defendit numerus
From: Ross A. Finlayson on 16 Oct 2006 21:35 Virgil wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > I warned you that this point is new: The edges are split in shares of > > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > > most people may have had the same problems as you today. I am sure you > > can understand it from the written text above. (Many others have > > already understood it.) > > Name one. Oh, don't worry, that's a problem everywhere. Consider the differences between for any, for each, for all, and for every. They're normally lumped under interchangeable representations of the universal quantifier. The transfer principle doesn't always hold. Yeah, like some years ago there weren't methods to solve the wave equation. Steve: It is. Alan: deperdit numerus. Consider the chicken farm. They're certainly well-fed. Have a nice day, idioten. It's in the null axiom theory. Ross
From: Dik T. Winter on 16 Oct 2006 21:59 In article <1160933634.254984.13910(a)e3g2000cwe.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Dik T. Winter schreef: > > In article <1160856043.795135.198610(a)h48g2000cwc.googlegroups.com> > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > > > > (BTW, I find > > > Banach Spaces not very useful either) > > > > They are (as far as I know) used in the design of methods to solve partial > > differential equations. > > Right. They are supposed to be basic for the Finite Element Method, to > be precise. But I have done quite some Finite Elements myself, and my > judgement is that they are not very basic. Here you can read what stuff > IMO is basic: Upfront I will state that I do not know much of this stuff. But: > http://hdebruijn.soo.dto.tudelft.nl/jaar2004/purified.pdf I found the statement at the bottom of page 3 quite interesting. You give two equations and you state that: "So we have two representations, which are almost trivially equivalent" In what way are they *not* trivially equivalent? > http://www.xs4all.nl/~westy31/Electric.html#Irregular > http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/belgisch.pdf But from those articles I can not find what you think is basic. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 16 Oct 2006 22:01
In article <290c1$45333e14$82a1e228$8972(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Dik T. Winter wrote: > > > In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > > Virgil schreef: > > ... > > > > I do not object to the constraints of the mathematics of physics when > > > > doing physics, but why should I be so constrained when not doing physics? > > > > > > Because (empirical) physics is an absolute guarantee for consistency? > > > > Can you prove that? > > Is it possible to live in a (physical) world that is inconsistent? We do not live in a world that is empirical physical. You previously talked about the world of approximations, where sqrt(2) would be rational. It is easy to prove that *that* world is inconsistent. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |