An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Virgil on 17 Jul 2006 16:39 In article <1153143649.224271.26480(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. And "following after" does not imply it is a "successor" and more than 6 following after 3 implies that 6 is the successor to 3, except, say, among multiples of 3. > There is no number following all naturals and, hence, there is no > number omega at all. There is no such 'natural' but there is such a cardinal or ordinal. Both are generically 'numbers' not all cardinals or ordinals are naturals. > > > > One has the set of all naturals, and that set can have a successor under > > the definition that the successor of any set, x, is (x union {x}). > > That is nonsense too. One cannot have a set of all naturals, because > the step x U {x} defined by the axiom does never result in this set. No one said that the set f naturals was a successor, stupid, we only say that it has one. Misrepresentations like that are the tools of trolls. > You recently asserted that such stepwise processes can never exhaust a > set. Now you changed your mind? Just in time? One can have a set of all of an inexhaustable supply of objects even though one cannot exhaust that supply by serial operations. > > Regards, WM
From: mueckenh on 17 Jul 2006 16:42 Dik T. Winter schrieb: > > In German it is "Umformungen", that means "change of the order". Cantor > > says: those changes, which and only which can be traced back to > > finitely many or infinitely many transpositions of elements, each > > transposition including two elements. > > So you claim that Cantor meant unrestricted re-ordering. If he meant that, > his statement is false. That is entirely possible, of course. Of course. His other statements on infinity are wrong too. In particular that which says that 0.111... is different from every natural number but no digit is different from every digit of every natural number. > > Now you use an entirely new term: "can be exhausted". I think you mean > that you can take out elements one by one and doing this at some stage > the infinite set becomes empty. Howver, I think, that if that can be > done, that there is a last element you can take out. And, according > to the axiom of infinity, that is not possible, so infinite sets can > not be exhausted in this sense. But in another sense? > > > That's all which is necessary to disprove set theory. > > You use a contradiction of the axiom of infinity (can be exhausted), so > there is no disproof. Now try the same with the assumptions that > infinite sets do exist and can not be exhausted. Then there is no evidence that they exist. It is the same as with God. There is no interaction with reality or with mathematics unless you believe in this interaction. But if there is interaction with an infinite set, then this set is easily proved to be not existing. The axiom of infinity does nothing else but o reflect and to formalize Cantor's position. But his position was also, that infinite sets can be exhausted if for each element there is a precise definition like that leading to the well-order of the rationals. The well order of the ordinals is another example. And the transformations consisting of transpositions is a third. > > > > > > Yes, that is a subset of the Umformungen. What is his definition of that > > > > > term? What does he mean when he writes "Umformungen"? > > > > > > > > I you are you unable to read English as well as German, try to get a > > > > copy in Dutch. > > > > > > Pray complete for me: transformations are ... > > > > Only "such transformations" need be defined. Such transformations are > > changes of the order which can be accomplished by transpositions > > (interchanges) of two elements. > > Well, I think *any* re-ordering can be accomplished by a sequence of > transpositions of two elements. Cantor shared your opinion, but only in case of finite sets. > > But when I write: > The question, though which froobles the ordinal number of a > well-ordered set is changed, and through with it is not changed, is > easily answered, those, and only those froobles leave the ordinal > number unchanged that can be rewritten as a finite or infinite > set of transpositions, that is, of interchanges of two elements. > you do not need the definition of froobles? I can tell you that my > statement is entirely correct. (I can state that your sequence of > transpositions does not form a frooble.) If you exchange "can be rewritten as" by "can be traced back to" or by "consist of", as is the meaning of Cantor's German statement, then "those froobles" is obviously merely another name for " fin. or inf. set of transpositions". > > > > There is no proof necessary. If infinite sets exist and can be > > > > exhausted, then the set of transpositions can be finished. As no single > > > > transposition changes the ordinal number, it remains unchanged during > > > > the whole proess.. > > > > > > Ah, but you have here an additional qualifier: "can be exhausted". Well, > > > infinite exists (the axiom of infinity), but can not be exhausted with > > > extraction of one element each time. So now what? You need a proof. > > > > If they cannot be exhausted, then they do not exist actually but only > > potentially. Then they have no cardinality. > > Eh? Infinite sets do exist, but they can not be exhausted. So why do they > not have cardinality? It is entirely possible to define cardinality for > them. What is your problem with that? That it is not possible to do so without well-ordering all elements. That is nothing but an exhaustion. > > One should have seen that earlier, then Bourbaki would not have > > succedeed to define 0 as natural number, even in political decisions. > > Pray explain the last part "even in political decisions". It is laid down in the guide lines of the European Community that zero was a natural number. I am indebted to your compatriots that they have dismissed the constitution of this disastrous association. >But I can rewrite > it in non-Bourbaki: > Reorder the naturals to (2, 3, 4, ..., 1) and see that the order type is > now w+1, so it did change from w. That would be nice, if omega could be exhausted. But it cannot. Hence, there is no omega + 1. > > You are fighting the mathematical use of limits. On what basis do you > allow the computational use? On practical basis. As precise as any computation can be the usual definitions are correct. Regrds, WM
From: mueckenh on 17 Jul 2006 16:51 Dik T. Winter schrieb: > > 0 was invented less than 2000 years ago. Natural numbers need not be > > invented. However, Bourbaki and Halmos tried this trick in order to > > prevent set theory from too easily been demasked as inconsistent. > > Eh? I was not talking about invention but about definition. What problems > do you have with definitions? I don't like definitions which define nonsense like the corner of a circle. > Yes, it was sloppy terminology. What happens when n grows without bound? Nothing happens with the *+ sum. > What is lim{n -> oo} SUM{i = 1 .. n} A_i ? How do you define that? > Without such a definition I have no idea what the result is when I *+ > all An. If there is at least one 1 in a column then the *+sum is 1. You need not investigate how many 1's are there to follow if you want to calculate the *+ sum. > > > > > But it is a unary representation of a whole number, namely of the > > > > number of digits of a real number. Nevertheless, in decimal > > > > number of digits of a real number. Nevertheless, in decimal > > > > representation 0.111... does exist, but not in the list. The addition > > > > of the list numbers gives the largest number of the list. > > > > > > In the finite case. As you have not provided a way to do the infinite > > > sum, I have no idea what happens in that case. > > > > 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. > 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. > Why than do you write that it is false? Because it is not correct. According to the axiom of infinity [...] infinite sets can not be exhausted in this sense. (Dik T. Winter) Therefore, a unary representation of a natural number can never reach the line next to the unary representation of aleph_0. The latter does exist according to set theory. The natural next to it does not. Hence, the sum of 0.1 0.11 0.111 .... is *not* 0.111... If exhaustion were possible, then I offered the natural oder by size and simultaneoulsy the well-order of the rationals. THERE IS NOT THE FAINTEST DIFFERENCE. Regards, WM
From: mueckenh on 17 Jul 2006 16:56 Franziska Neugebauer schrieb: > > It reads: Either there is a column with only zeros, or there is at > > least one 1 in each column spanned by the digit positions of 0.111... > > , isn't it? > > But you will "argue": Nice try but there is neither nor. > > What you call a proof is an unproven proposition. IF aleph_0 does exist, THEN 0.111... covers aleph_0 columns. IF 0.111... covers aleph_0 columns, THEN aleph_0 columns do exist. IF aleph_0 columns do exist THEN we can consider their contents. IF we can consider the contents of each column, THEN we can ask how many 1's are therein. 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". IF the answer is in each case is "not zero 1's", THEN in each column at least one 1 must be present. However, there is no natural numbers with this property, because 0,111... has more 1's than each natural number. Hence 0.111... itself must be present among its disciples. Unheard and unseen but in the midst among them. ----- Matheology. > >> > Is it in particular point 4 which excites your fury? > >> > >> I will not read it. > > > > The ultimate defense. > > Self protection. Such public confessions are unusual in matheology. Regards, WM
From: Virgil on 17 Jul 2006 17:14
In article <1153144813.275459.154630(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Hartley schrieb: > > > >> >> I cannot comment on the German text, but Dik's reading of the English > > >> >> is > > >> >> clearly correct, WM's wrong. To put it even more clearly > > >> >> > > >> >> Question: Which transformations preserve ordinal number? > > >> >> > Question: What is an order-preserving transformation? A function f: A --> B such that f(x) > f(y) if and only if x > y. But that is not the issue here. It is the well-ordering of the set , not its ordinary ordering that is to be preserved, and any transpostion will destroy the original ordering, but not the well ordering. > > > > As I and others keep reminding you, it doesn't make any difference what > > Cantor actually meant. Quoting Cantor does not constitute proof. > > But outspoken wrong interpretations like that of Dik and yours must be > corrected. Since you seem to object violently to having your own mistakes corrected, one does not see that your attitude is at all justifiable. > > >If you > > want to make your supposed proof of inconsistency rigorous, you must: > > > > a) define what you mean by the result of applying an infinite sequence > > of transpositions, > > Has been done. Not to the satisfaction of anyone else, it hasn't. > > > > b) prove that your particular sequence re-orders a well-ordering of the > > positive rationals to the usual ordering, > > > > c) prove that such a sequence cannot alter the order-type. > > It is impossible to have a dense order and a well order simultaneously. Then you cannot do what you claim to have done. > > > > (Naturally, you will do this within a standard formulation of modern set > > theory.) > > > > Dik and I think you can do b but not c. Virgil thinks you can't do b, > > but until you provide a your claims are "not even wrong", just > > meaningless. > > What would a proof mean? Dik would think this and Virgil would think > that and you would perhaps have another opinion, but nobody would > accept this as a proof, because all like set theory and then it would > be dead. So far WM has provided no proof that is valid within any standard set theory, i.e., that does not require assumptions not a part of ZF or NBG. > > Set theory is meaningless, but it has the advantage that it cannot be > wrong. Which gives it a tremendous advantage over WM, who can rarely be right. |