An uncountable countable set
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From: Dik T. Winter on 18 Jul 2006 20:23 In article <1153242462.523960.281130(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1153148551.942037.97110(a)s13g2000cwa.googlegroups.com> muecken= > h(a)rz.fh-augsburg.de writes: > > > > > Dik T. Winter schrieb: > > ... > > > > (1): 0.111... is not *the* successor of anything, we may sloppily > > > > say that it is *a* successor of all naturals, just like 10 is > > > > *a* successor of 2 > > > > > > It was Cantor who coined this expression saying " da? omega die erste > > > ganze Zahl sein soll, welche auf alle Zahlen nu folgt." (Works, p. > > > 195) So it must be larger anyhow. > > > > Yes. Not *the* successor, but *a* successor. > > The "first" successor. Tere is only one "first" successor! Conveniently snipping your own text again? You wrote in <"news:1153052131.852951.273540(a)b28g2000cwb.googlegroups.com">: > Hence, > 0.111... as the succesor of all naturals must consist of more 1's, than > any natural, if it is to be a number. Now, if you had written "the first successor" I would not have made any remarks. It is not *the* successor. > > Are you again arguing that the statement > > For all p, there is an n such that An[p] = K[p] > > That statement is false. (Without K including the index p = aleph_0, k > cannt be different from all natural numbers.) What has your statement in parentheses to do with the statement I gave? There is no argument at all about K being different from all natural numbers. It is just a plain statement: For all p, there is an n such that An[p] = K[p] you claim that is false, and again without direct proof. Do you disagree with the statement: For all p, Ap[p] = K[p]? I know that Ap[p] = 1 = K[p]. Now chose in the above n = p and you get the new statement. Why would this second statement be false? > > is false, or are you arguing that the statement > > For all p, there is an n such that An[p] != K[p] > > is false? > > That statement in the form An != K is correct. The seqence An[p] does > not exist for all p which are in K. Again, you do not answer the question. Do you disagree with the statement For all p, Ap-1[p] != K[p]? I know that Ap-1[p] = 0 != 1 = K[p] (I see now that p>1 is a requirement). Now chose in the above n=p-1 and you get this statement. What is false about this statement? > > > There is no actually infinite set of finite numbers. > > > > Axiom of infinity. > > In contradiction with mathematics, obviously. You have not yet shown it. Only in contradiction with *your view* of mathematics. That is something entirely different. -- 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 18 Jul 2006 20:49 In article <1153243255.293378.172400(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1153169497.380974.32190(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > I don't like definitions which define nonsense like the corner of a > > > circle. > > > > Check the manhattan-measure and you will find square circles. > > Then I'll better do without. Yes, you dislike definitions that do not conform with your views. You are not interested in the possible usability. > > That is not an answer to my question. But I will take it in good faith, > > so > > lim{n -> oo} SUM{i = 1 .. n} A_i = 0.111... > > tell me where I have gone wrong. > > You cannot exhaust the naturals. Therefore what you write as 0.111... > is not the same as the unary representation which also is denoted by > 0.111... but means that N is exhausted and there is at least one 1 > which cannot be indexed by a natural number. Makes no sense to me. I see no exhaustion, see, I did use the symbol "lim" there, and I had the hope that there would be a proper definition. Apparently not. > The reason is that > > 0.1 > 0.11 > 0.111 > ... > gives allegedly the same sum as > > 0.111... (as representation of aleph_0) > 0.1 > 0.11 > 0.111 > ... Yes, with a proper definition of "...". What was the reason again? You apparently have another definition in mind. Pray give *your* definition. > > > > > Define: If any case includes at least one 1 the the *+ sum is 1. > > > > > > > > Again, in the finite case. You have not defined what you mean with the > > > > infinite sum. > > > > > > Hell and devil! Can't you read? The definition for *+ sum = 1 is: at > > > least one 1 must be encountered. That is enough in any case, finite or > > > not. > > > > Damnation leaving aside, you defined what SUM{i = 1 .. n} An is. You did > > *not* define what happens when n grows without bound. Now you state that > > in each column, whenever there is a 1, the final result should be 1. > > Here is another example for the *+ sum: I do not ask for examples, I ask for definitions. And those you refuse to supply. Now clearly above I did think about definitions, and used them and you state I was wrong. Without even giving the correct definitions. > > But > > what you are now meaning is that > > lim{n -> oo} SUM{i = 1 .. n} An = 0.111... . > > I said above: It is different from n really reaching infinity. But for > 1/9 or aleph_0 infinity is reached. With a proper definition of the limit, the sum will *never* reach 1/9 (or aleph_0). But as you still refrain from giving definitions of the notations you are using this discussion is less than fruitfull. > > > > So the statement > > > > For all n, An[n] = K[n] > > > > is true? As is the statement > > > > For all p there is an n such that An[p] = K[p] > > > > also true? > > > > > > No. Then 0.111... would not differ from every n. > > > > No to which question? Is the first statement false? And if so, why? > > Is the second question false, and if so why? And how do you come at > > your conclusion? > > The reason is that I ask questions. Again you refrain from giving answers. Are you unwilling to give answers? > 0.1 > 0.11 > 0.111 > ... As my interpretation of your "..." was apparently wrong above, pray, finally supply a definition of that notation. If you do not supply that there is no reason to discuss any further, as you are not willing to provide enough information about what you are writing. > > You are arguing on two different lines at the same time, confusing one > > with the other. When we consider the An as unary representations of > > natural numbers the result is aleph_0, not a natural number. When we > > consider them as being decimal numbers, the result is 1/9. > > How many digits has 1/9 in decimal representation? That is not a natural number. > > > 0.1 > > > 0.11 > > > 0.111 > > > ... > > > is *not* 0.111... > > > > With your definition (finally extracted) above (if in any column there is > > a 1, there is a 1 in the final result), it is. If you think that is false, > > please show me a column where there are only 0's. > > That column which cannot be covered by a natural number. > (f all couldcovered then 0.111... was a natural.) There is no such column. And 0.111... is not a natural. What is in *your* opinion the result of *+-ing all natural numbers togther? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 19 Jul 2006 05:38 Dik T. Winter schrieb: > > If the *+ sum of the list is a unary representation of some 1's, then, > > by the construction of the unary numbers of the list, this sum must be > > a unary numer of the list. > > As long as your list is finite. As long as it contains *numbers* which can be distinguised from each other and which obey trichotomy. > Suppose the list is infinite, and the > *+ sum of all members of the list is a member of the list. That would > mean that the *+ sum of the list is the last element of the list, > contradicting that the list is infinite. If we assume that non-terminating fractions have aleph_0 digits, then this case is realized by 0.1 0.11 0.111 .... 0.111... Here the last number of an infinite list is in this list. Actual infinity or, in other words, finished infinity is the necessary consequence of the set-theoretical theorem that infinity can be surpassed. > Hence the assumption is false. > -- Yes, the assumption is clearly false that omega or aleph_0 is a number larger than any natural but counting all the naturals. The assumption is false that irrational numbers and non terminating fractions have as many digits as there are natural numbers *and* that this is a number aleph_0. Regards, WM
From: mueckenh on 19 Jul 2006 05:49 Virgil schrieb: > > > > > > The successor of all naturals is not a natural and, therefore, must be > > > > > > larger (because it is not less). > > > > > > > > > > There is no such thing as the successor of all naturals any more that > > > > > there is a single successor common to both 3 a and 6. > > > > > > > > This is an expression coined by Cantor: "a number following after all > > > > natural numbers". > > > > > > But he does not call it a 'natural' number any more than he calls it an > > > even number or a prime number. > > > > I said above: The successor of all naturals is not a natural. > > No one else even says that such a thing exists. There is a set of all > naturals but if is not the successor of any of them. Cantor said it. And if it is a number, then it must e lager or equal or less than a natural number. As he latter is impossible, it can only be larger. > > > > > > There is no number following all naturals and, hence, there is no > > > > number omega at all. > > Perhaps not in your philosophy, but you are not God, to command what is > true or false. But I can see what is contradictory. > > > > > > There is no such 'natural' but there is such a cardinal or ordinal. Both > > > are generically 'numbers' not all cardinals or ordinals are naturals. > > > > Why do you emphasize this? Of course it cannot be a natural if it > > follows after all naturals. > > I emphasize it because WM needs to learn it. You are a liar. Therefore I cease my correpondence with you. Last regards, WM
From: mueckenh on 19 Jul 2006 09:16
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > [...] > > >> > IF we can ask how many 1's are in each one, THEN the answer can be > >> > "zero 1's" or "not zero 1's". > >> > >> We can. > >> > >> > IF the answer is in each case is "not zero 1's", THEN in each > >> > column at least one 1 must be present. > >> > >> This is the case, since every a_jj = 1 j e N by definition. > >> > >> > However, there is no natural numbers with this property, > >> > >> Could you precicely _define_ which /property/ you are talking about? > > > > To contribute a 1 to each column. > > Should be a property of which object? Should be a property of at least one summand if the *+ sum has this property. > > >> For every column j e N a_mj has the 1 in position m(j) = j, since > >> a_jj = 1 A j e N. Where exactly lies your problem? > > > > I told you recently: In a list like mine a *+ sum is defined. This * > > sum is equal to the largest number of the list. > > The notion of "largest number" is misleading. There is no largest > number. Neither in the list nor in omega. If it makes sense to talk > about "a largest number", then please 1) That does not mean that your pretension (sum of all naturals is 0.111...) was correct.2) omega is the largest number less than omega + 1. > > 1. name it (show us its name or position _in_ the list or _in_ omega), > and/or > 2. present a _proof_ that it exists. In 0.1 0.11 0.111 .... 0.111... this element exists , if it does exist at all. Its position can be selected arbitraily. > Until then we cannot (meaningfully) talk about "a largest number in the > list" or "a largest number in omega". Wittgenstein applies here. omega is the largest number of he sum above. Its existence is proven in set theory. > > > And if a larger number is included, then the sum grows. You pretend > > that this is not valid for 0.111... because without 0.111... the list > > has the *+ sum 0.111... and when you include it in the list (as > > diagonal number or as the first line or anywhere else) the *+ sum does > > not grow. > > Postponed until existential status of "largest number" is clarified by Is clarified. Regards, WM |