An uncountable countable set
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From: Han de Bruijn on 29 Sep 2006 04:17 stephen(a)nomail.com wrote: > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > >>Virgil wrote: > >>>In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Randy Poe wrote, about the Balls in a Vase problem: >>>> >>>>>It definitely empties, since every ball you put in is >>>>>later taken out. >>>> >>>>And _that_ individual calls himself a physicist? >>> >>>Does Han claim that there is any ball put in that is not taken out? > >>Nonsense question. Noon doesn't exist in this problem. > > Yes it is a nonsense question, in the sense > that it is non-physical. You cannot actually perform > the "experiment". Just as choosing a number uniformly > from the set of all naturals is a non-physical nonsense > question. You cannot perform that experiment either. But you _can_ do it at any time _before_ noon. There is no limit of the number of balls before noon which converges at noon. But you _can_ do it with any finite contiguous set of naturals. Then you find floor(n/a)/n and with limit(n -> oo) find 1/a . Han de Bruijn
From: Dik T. Winter on 29 Sep 2006 07:44 In article <97d49$451cd1e0$82a1e228$15818(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Virgil wrote: .... > > Def, corected: 0.333... = lim_{n -> oo} Sum_{k = 1..n} 3/10^k > > > > And it exists because there is a real number (actually a ratonal number) > > L = 1/3 such that for every epsilon greater than 0, there is a largest > > n such that | L - sum_{k+1..n}| >= epsilon. > > Shouldn't that be "such that | L - sum_{k = 1..n}| <= epsilon" ? And why > that "largest n" instead of just "n" ? No to both questions. It there is a largest n such that |expr(n)| >= epsilon, then for all m > n |expr(m)| < epsilon. Just what is needed. -- 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 29 Sep 2006 09:16 In article <1159437621.889544.5040(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > > Because there is only one set that contains *all* natural numbers. > > > > > > Why? Your assertion is without proof. > > > > I should have stated: "there is only one set that contains *all* natural > > numbers and no other numbers". But that is easily proven once you > > understand set theory. > > And you will easily see that this proof is rubbish once you understand > a little more than set theory. Oh. What should be understood? > "All natural numbers and some more > natural numbers" is the same as "all natural numbers". What is "some more natural numbers" > For instance: > Does the set o all natural numbers include 0? In old Greek it did not > even include 1. In future it may include even -1. Yes, indeed, that depends on your starting point with the natural numbers. That does not make it "all natural numbers and some more natural numbers". > And in all these > versions you insist that there be only one number 0.111...? Yes, indeed. Because there is a bijection between the various sets of natural numbers when you use different definitions. When the naturals start with '1', the '1' immediately after the period is digit number '1' after the period. When the naturals start with '0', it will be digit number '0'. > > > Why should there be only one number > > > 0.111... ? By what property is this 0.111... different from all the > > > numbers in the list? > > > > Because it has infinitely many digits? > > That is not a property which can distinguish it from the numbers of my > list. Why not? Each and every number of the list terminates. That one is a number that does *not* terminate. > If you think that 0.111... is a number, but not in the list, It is *you* who insists it is a number. In most of my communications with you I put the word number in quotes, because it depends on how you interprete it on whether it is a number or not. > then > you must be able to find a position which is different from those of > the numbers of my list. No. It is sufficient to prove that for each number on your list there is a position where it is different from that number. That does *not* imply that there is a position where it is different from each number of the list. > > > And why can't there be more than one number with > > > infinitely many digits? You cannot answer these questions because > > > already one infinite set is a contradiction. > > > > No, I can not answer this question because I have no idea what you mean > > with a number with more than omega digits. Consider K = 0.111... . What > > is K+1? Can you provide a definition (as I did for K)? > > k + omega is omega. And -k + omega is omega too. There is no well > defined set. In what way is that an answer to my question? Do you understand that 1 + omega = omega != omega + 1? And (as far as I know) -k + omega is not defined for positive k. (With the ordinals addition is defined only between ordinals.) > > > Which index distinguishes 0.111... from all the numbers > > > of the list? You cannot answer? > > > > I can. None. > > The axiom extensionality tells us that two sets are different, if they > differ in at least one element. If 0.111... differs from number n, then > it differs from all numbers m < n. As 0.111... is different from each > number of the list, it also differs from each one which is smaller than > another one. As every number of the list is smaller than another one, > 0.111... cannot be covered by all numbers. Hence, it cannot be indexed > by all list numbers. Again, that last conclusion is not justified. > > So the number can be indexed. > > It is curious. You could also assert something like "I can shout louder > than anybody else but there is nobody who cannot shout louder than me". > But it is impossible to try to exorcise your burnt-in anti-logicism. Apparently not with your burnt-in anti-logicism. I have *proven* that it can be indexed, by the simplest of all possible proofs. Namely by showing that there is no digit 1 at any position other than indicated by a natural number, which *by your* definition makes the number indexable. > > > So we cannot answer which index > > > distinguishes the many different infinite digit sequences 0.111... from > > > each other. > > > > What different infinite digit sequences? I note that digit sequences are > > countable, and so there is only one infinite digit sequence. > > I note that some sequences are finite, and so there is only one finite > digit sequence? > Is that really an argument? Not for finite sequences, but it is for infinite sequences. > > > Either: There is an index which distinguishes 0.111... from any number > > > of the list. > > > Or: There is a number which cannot be distinguished by indexes. > > > But if there was one such number admitted, how could the existence of > > > many of them be excluded? > > > > Because there is a single minimal countable but infinite set? > > The set with twice (or half) as many elements is not called a minimal > countable but infinite set? Minimal for sets that start with 1 and contain all the successors of the elements in the set. -- 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 29 Sep 2006 09:20 In article <1159437845.922031.117160(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > For instance: I + I = II. > > > > In some cases, yes. In other cases, no. Depends entirely on how you > > define "+" and the other symbols. In Greek mathematics I would expect > > I + I = K (as in 10 + 10 = 20). .... > > I ask for self-evident truths. Upto now you have not provided any. > > For Cantor I + I = II is such a self-evident truth. (Of course with the > usual meaning of "+" and "=".) How do you *define* "+"? I have not yet seen a definition from you. In mathematics that operation can be defined in terms of the Peano axioms. Moreover, in some rings 1 + 1 can be something entirely different, like 0. (Take any field of characteristic 2.) > For me too. Or take another one: If you > divide a sphere in a few parts and afterwards put them together again, > then you will get one sphere and not two. Yes, you think some conclusions from mathematics are ridiculous, so you think there must be an inconsistency. But the inconsistency is there only with respect to your expectations, not internally. -- 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 29 Sep 2006 09:26
In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > The successor function *is* counting (+1). > > > > Wrong. > > After a while you will have run out of the predefined successor, > unavoidably. succ(x) = {x}. > Then you have no other choice but to add 1 each time you > proceed. That is counting. Take the reals. succ(r) = sqrt(1 + r^2). Where is the counting? > > > The successors are defined > > > without counting only over a very restricted domain. In the usual > > > decimal systems only from 1 to 12 and then repeating again and again > > > from X to X + 9. > > > > You think so because you again focus on the decimal system. I wonder how > > you get at 12. German influence? > > Of course, but not only in German or English we see that phenomenon. I > would be surprised if it were different in Dutch. Germanic languages in general. In Indian languages in general to 99. In Welsh to 20. In Yoruba most similar to from -4 to 5. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |