An uncountable countable set
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From: Mike Kelly on 19 Sep 2006 10:23 Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >> > >>>Han de Bruijn wrote: > >>> > >>> > >>>>Mike Kelly wrote: > >>>> > >>>> > >>>>>Han de Bruijn wrote: > >>>>> > >>>>> > >>>>>>Mike Kelly wrote: > >>>>>> > >>>>>> > >>>>>>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>>>>>> > >>>>>>> > >>>>>>>>Mike Kelly wrote: > >>>>>>>> > >>>>>>>> > >>>>>>>>>Infinite natural numbers. Tish and tosh. Good luck explaining that idea > >>>>>>>>>to schoolkids. > >>>>>>>> > >>>>>>>>Look who is talking. Good luck explaining alpha_0 to schoolkids. > >>>>>>> > >>>>>>>Sure, the theory of infinite cardinals is beyond (most)schoolkids. But > >>>>>>>this is a bad analogy, because school kids don't need to know about > >>>>>>>cardinals but they do need to know how to work with natural numbers. My > >>>>>>>point, if you really missed it, was that Tony's ideas of "infinite > >>>>>>>natural numbers" don't match up to our "naive" or "intuitive" idea of > >>>>>>>what numbers should be - how we were taught to do arithmetic in school. > >>>>>>>I for one don't understand what the hell an "infinite natural number" > >>>>>>>is. And yet supposedly the advantage of his ideas are that they're more > >>>>>>>intuitive than a standard formal treatment. > >>>>>> > >>>>>>My point is that the pot is telling the kettle that it's black (: de pot > >>>>>>verwijt de ketel dat ie zwart is). Your aleph_0 is in no way better than > >>>>>>Tony's "infinite natural number". > >>>>> > >>>>>Your analogy is terrible, as usual. > >>>>> > >>>>>My point was that Tony's "infinite natural numbers" are not compliant > >>>>>with everyday arithmetic. Aleph_0 is part of a formalisation that leads > >>>>>to an arithmetic that works exactly as we expect it to. > >>>> > >>>>"... that works exactly as we expect it to". Ha, ha. Don't be silly! > >>> > >>>So, what part of the arithmetic on natural numbers defined rigorously > >>>as sets doesn't match up to the "naive" arithmetic we were taught at > >>>school? > >> > >>I thought you meant the arithmetic with transfinite numbers. No? > > > > > > In what way is the arithmetic of transfinite numbers part of everyday > > arithmetic??? > > Precisely! What the hell are you talking about? Arguing with someone who can't speak English is getting aggravating. I claim that Aleph_0 is part of a formalisation that leads to an arithmetic on natural numbers that works just how naive arithmetic works. Do you disagree? -- mike.
From: Mike Kelly on 19 Sep 2006 10:24 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > > > All naturals do not exist. What is "all"? > > > > Huh? So some naturals don't exist? What does that mean? How can > > something that doesn't exist be a natural number? > > Not all novels do exist. Is that meaning clear to you? Not in the slightest. > > You are being dishonest by misrepresenting what people say to support > > your position. > > > Isn't that sentence applicable to your above question too? How is a request for clarification of broken English a misrepresentation? -- mike.
From: Mike Kelly on 19 Sep 2006 10:31 mueckenh(a)rz.fh-augsburg.de wrote: > Mike Kelly schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Mike Kelly schrieb: > > > > > > > > Any set that can be established is a finite set. > > > > > > > > Why? > > > > > > Look: If aleph_0 were a number larger than any natural number, then for > > > any natural number n we had n < aleph_0. "For all" means: even in the > > > limit. > > > > OK so far. Every cardinal number which is a natural number is less than > > aleph_0. > > > > > So lim [n-->oo] n/aleph_0 < 1 > > > > Division is not defined for infinite cardinal numbers. > > Is that your only escape? If you dare to say that aleph_0 > n for any > n e N, then we can conclude the above inequality. No, because division is not defined on infinite cardinal numbers. The above inequality is meaningless. >But remedy is easy. > Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions > analogously. > > > > > If aleph_0 counted the numbers, for instance the even naturals, then we > > > had for all of them > > > > > > lim [n-->oo] |{2,4,6,...,2n}| = aleph_0. > > > > > > This would yield lim [n-->oo] (2n/|{2,4,6,...,2n}|) < 1 > > > > > > But we have lim [n-->oo] (2n/|{2,4,6,...,2n}|) = 2 > > > > > > Therefore aleph_0 does not exist as a number which could be compared > > > with other numbers. > > > > > > > > > No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > > > > > least as close to 2 as we like), not by definition and not by any > > > > > axiom, but by rational thought. > > > > > > > > Prove that to be the case without using any definition of what a series > > > > is and without any axioms. > > > > > > Archimedes did so when exhausting the area of the parabola. In decimal > > > notation 2 + 2 = 4, and in any system we have II and II = IIII. > > > > In airthmetic modulo 3, 2+2 = 1. > > If you say "in arithemtic mod 3", then you imply that you subtract 3 > from the true result as often as possible. It does not invalidate II + > II = IIII, if you subsequently tale off III. Huh? The "true" result is that 2+2 = 1, if you are working in arithmetic modulo 3. Or if it's 10 o'clock now and I wait 5 hours then it is 3 o'clock. Your position seems very inconsistent. You claim that numbers have no existence outside their representation. And now you are claiming there exists a "true" arithmetic. > > >For self-evident truths you don't need axioms. Only if you want to > > > establish uncertain things like "There exist a set which contains O and > > > with a also {a}" then axioms may be required. > > > > > > Don't misunderstand me: I do not oppose the principle of induction but > > > the phrase "there exists" which suggests the existence of the completed > > > set. > > > > Why do you object to this? > > Because of he proof above. It is not a proof. Division is not defined where either operand is an infinite cardinal number. -- mike.
From: Han de Bruijn on 19 Sep 2006 10:39 Mike Kelly wrote: > What is "active mathematics"? > I'm not really seeing the point in this line of discussion. In what way > does any of this dispute that probabilities are not infinitesimal? > I'm not attempting to argue in Dutch, am I? Your attitude is quite constructive, huh? Han de Bruijn
From: Han de Bruijn on 19 Sep 2006 10:44
Mike Kelly wrote: > What the hell are you talking about? Arguing with someone who can't > speak English is getting aggravating. My English is much better than your Dutch. > I claim that Aleph_0 is part of a formalisation that leads to an > arithmetic on natural numbers that works just how naive arithmetic > works. Do you disagree? I disagree. Aleph_0 does NOT lead to a blah blah arithmetic on natural numbers. The natural arithmetic existed long before aleph_0 was born. Han de Bruijn |