An uncountable countable set
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From: Han de Bruijn on 19 Sep 2006 07:22 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! Han de Bruijn
From: Han de Bruijn on 19 Sep 2006 07:23 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 de Bruijn wrote: >>>>>>> >>>>>>> >>>>>>>>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? >>>>>> >>>>>>"All naturals" is undefined, void of meaning. Got it? >>>>> >>>>>Not really, no. If something is true for "All naturals" it is true for >>>>>any object which is a natural number. The set of all naturals is the >>>>>set which has as an element every object which is a natural number and >>>>>no element which is not a natural number. >>>>> >>>>>I suppose you're just going to respond "but there is no such set!" - >>>>>presupposing your conclusion; that there are no "completed infinities". >>>>>Vigorous assertion is so convincing. >>>> >>>>So there is no lack of understanding on your part. Good! >>> >>>So you admit you don't actually have an argument beyond assertion? At >>>least mathematicians recognise that the "existence of completed >>>infinities" is not provable one way or the other. >> >>No argument beyond assertion AND (empirical) scientific evidence: >> >>http://huizen.dto.tudelft.nl/deBruijn/grondig/natural.htm#oo > > What bearing does empirical evidence have on what is mathematically > meaningful? How serious is mathematics? Han de Bruijn
From: mueckenh on 19 Sep 2006 07:28 Mike Kelly schrieb: > > Please make just the experiment. Choose at random 30 natural numbers > > from the whole set N. What is the result? How many of these 30 numbers > > are in fact divisible by 3? (In case you have problems with large > > numbers: It is easy to check the divisibility of the number by checking > > the divisibility of the sum of its decimal digits.) > > > > Now, it there a distribution lacking, or is the complete set of natural > > numbers lacking? > > Neither. There is a lack of distribution *for the complete set*. If here is the set, than you do not need a distribution. If you have access to each number, then select just blind by random choice. > > > > If you don't think this claim means anything at all then > > > why do you dispute it? If you reject the existence of the set of > > > natural numbers then you reject the set theory probability is based on. > > > > In order to calculate probability we do not need set theory. Pascal and > > Fermat, for instance, did it without set theory very well. > > But they did not use a rigorous probability theory. And a rigorous > theory becomes a necessity when dealing with probabilities and > statistics beyond the trivial. They used rigorous mathematics and they obtained correct results. Do you realy think that set theory made anything more rigorous? Do you think mathematical life will end when a contradicition in ZF or ZFC will be discovered? On the contrary: "The notions of set theory are relative. As a matter of fact, the absoluteness of the notions of number theory and geometry is to some extent deceptive. Those notions are based on set-theoretical notions, and since the latter turn out to be relative, the former are relative too" v. Neumann, 1925. > > >Your result > > shows only that set theory is not useful in any branch of useful > > mathematics. > > Hard to take you seriously when you say this. If it isn't useful, why > is it so widespread? Conspiracy? Force of habit? There are many things widespread which are not useful at all. (Look at what people believe.) > > > > If you refuse the idea of infinite sets, what does it mean to you to > > > say a function has domain and range R? > > > > As an argument you can choose any real which you really can choose. > > > > See the experiment above. You don't really believe that you can choose > > a natural from the whole set N, do you? > > Sure I can. I choose 7. But I didn't choose it uniformly at random from > all naturals. That is not from the whole set, but from its very first segments only. You see: There is a natural number n_0 which is larger than any natural number n which ever will be named, thought, chosen. From that part of the set which is larger than n_0 you can never choose a number. Regards, WM
From: Mike Kelly on 19 Sep 2006 07:29 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. > 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. > > > > > Hence, the probability to select a number divisible by 3 is 1/3 or very very close to 1/3. > > > > >From finite sets of consecutive naturals when selecting with a uniform > > distribution, sure. But you don't accept that there is a set of "all" > > natural numbers so what does it mean to you to select at random from > > "all" naturals? > > YOU should know it, if they all exist for you. Take all of them and > then select one without looking. OK, I choose 7. But I didn't do it uniformly at random. > > > > >Otherwise the > > > > > limit of the sequence 1/n might be 100. Nobody could prove that false. > > > > > > > > Babble. > > > > > > 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. >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? -- mike.
From: mueckenh on 19 Sep 2006 07:34
David R Tribble schrieb: > mueckenh wrote: > > "a reasonable way to make this conform to a platonistic point of view > > is to look at the universe of all sets not as a fixed entity but as an > > entity capable of 'growing', i.e. we are able to 'produce' bigger and > > bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set > > Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. > > Is there a sentence that follows that one, maybe about points of > view other than platonistic? Sorry, I am not in possession of this very good book. I noted only this very interesting sentence. But it should be clear, from the axiomatic point of view, we just do not think about the universe of all sets. Axiomaticans cannot be Platonists. Compare Goedel who wondered whether the continuum hypothesis is true or not in the real universe o sets. > > > Why should simple infinite sets exist in another way? Just because > > there is an axiom which cannot be satisfied like the axiom that there > > be a straight bent line? > > I assume you're talking about there being no set that satisfies the > Axiom of Infinity. Why can't there be such a set? There are several proofs. I just posted one to Mike Kelly. Regards, WM |