An uncountable countable set
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From: Dik T. Winter on 21 Sep 2006 08:17 In article <1158312563.101733.242910(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > Yes, it is not in the list. So what is the problem? All possible > > > > indexes are in the list, so: > > > > (1) A{p = digit position} E{q = list item} {such that q > > > > indexes p} > > > > which you deny. > > > > > > That is not an argument. > > > > *Why* is it not an argument? You state: "all possible indexes are in the > > list". What is *wrong* about the argument? You always state "it is > > wrong", but never state what part of the argument is wrong. > > It is wrong because not for all {p = digit position} there exists {q = > list item} {such that q indexes p} , but only for those p, which can > be indexed, i.e., which are present in the list. But see above: "All possible indexes are in the list". > As 0.111... is not in > the list, there must be a digit position, which is not in the list. (As > such a position cannot be identified, 0.111... cannot exist.) Again: define 0.111... is the string with an 1 in the p-th position for each natural number p, and with no other positions. I would think this is a clear definition. It clearly can be indexed. And there is not a digit position which is not in the list. *And it exists*. Your statement: "As 0.111... is not i n the list, there must be a digit position, which is not in the list" is unfounded and without proof. > > No. There are "numbers" (exactly one) that can be indexed by that list > > but that is not in that list. > > That is a false belief. And why should there be only one of those > numbers? Why couldn't it be two at least or even ten? If you answer > this question, you have a good chance to recognize that not even one > exists that is not in that list but can be indexed by the numbers of > that list. Because there is only one set that contains *all* natural numbers. > > But there is a single "number" that can be indexed by your list but not > > covered by your list. 0.111... is one (as I defined it above). But it > > is *not* a natural number. > > Therefore it cannot be indexed by natural numbers. Try to find out, why > you insist on only one of those strange numbers. There could be > infinitely many. Could they all be indexed by natural numbers?. There is only *one* set of natural numbers, so there is only *one* sequence 0.111... that can be indexed using *all* natural numbers. > > Do you agree that the non-terminating list can index non-terminating > > numbers? If the answer is no, why? > > Because it should index at least two different of those infinite > numbers if it could index one of them. Why? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on 21 Sep 2006 08:57 Han de Bruijn wrote: > stephen(a)nomail.com wrote: > >> You are in disagreement with WM, who has clearly stated >> that sqrt(2), pi, e and all the irrationals are not numbers. > > I think that you are babbling, but before saying so, I shall first read > what WM actually says about it. I can't find any document on VM's site where he says that sqrt(2), pi, e and all the irrationals are not numbers. Would you mind to point me to a reference? (I've tried "On the abundance of the irrational numbers") Han de Bruijn
From: Dik T. Winter on 21 Sep 2006 08:54 In article <1158313360.862711.19070(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > That does not exist. At least it is not described by n. III is a > > > representation of 3. > > > > Now you shift back to representation, pray remain consistent. > > Representation is number. There is no difference. Numerals have no > "soul". Makes no sense to me. You may represent a natural number as you wish, that does not change anything. > > > If you know the positions by heart, then you need no addition actually. > > > You had already used it or the person who devised that technique had > > > used it. But earlier or later your knowing by heart will end and you > > > will have to count +1. > > > > Nope. I will only to need to know successors. Anyhow, you read much more > > in the successor of the Peano axioms than is present. The successor is > > defined without even any knowledge of addition at all. So > > succ(George V) = Edward VIII, succ(Edward VIII) = George VI and > > succ(George VI) = Elizabeth II within the set of British kings and queens. > > I do not think what way of addition you would propose for that. Of > > course, this successor function does not satisfy all of Peano's axioms, > > but I hope you get the idea. > > I was talking of *counting* (remember: Zahl and zaehlen) which requires > natural numbers which require the ability to add 1 after you have run > out of the numbers known by heart. You stated that you needed counting to determine the successor. That is false. The successor is defined without any reference to counting. > > > The old Greek and othe cultures have just used their letters as numbers > > > too. > > > > Yes, every culture had their representation of numbers. Sometimes base 10, > > sometimes other bases. Base 20 is quite common in Europe. Mixed bases > > are also used. Sometimes they were used to count from 1. Sometimes from > > 0. > > Who did so before Cantor? Some people from India. There are quite a few where you can read about the zeroth year of the reign of somebody. Have a look at the vast number of different calendars and year countings that have been in use in India. > > I may note that the decimal system you appear to like most is, in > > Europe, mostly due to Simon Stevin, and at that time it included 0 as > > a digit. > > Did he count from 0? As he never wrote about it, I have no idea. -- 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 21 Sep 2006 09:07 In article <1158324386.661753.251440(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > Index = natural number. There is no infinite index, because indexing is > > > identifying. > > > > That does *not* explain why the list contains all numbers that can be > > indexed. > > Let m be a unary number with m 1's, and n a list number with n 1's. > If E n >= m, then m is in the list and can be indexed completely by > list numbers. > If not E n >= m, then m is not in the list and cannot be indexed > completely by list numbers. Yes. With finite m. But a unary "number" with infinitely many 1's does not fall under the statement. So it does not follow that it can not be indexed. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on 21 Sep 2006 09:19
Han de Bruijn wrote: > I can't find any document on VM's site where he says that sqrt(2), pi, e > and all the irrationals are not numbers. Would you mind to point me to a > reference? (I've tried "On the abundance of the irrational numbers") Oops! He _says_ it, in the document "Physical Constraints of Numbers": > Irrational numbers simply are not numbers. Weird ... Han de Bruijn |