Cantor Confusion
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From: mueckenh on 3 Nov 2006 07:10 William Hughes schrieb: > mueck...(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > Each diagonal digit is formed by a line. > > > > and by a column! > > Yes, and the number of columns is equal to the number of lines. > The fact that no single line contains all the columns is not > important. The fact that *no* set of lines contains all the line is not important too. I think we have obtained agreement, that we disagree in this point and that further discussion will not lead to a result acceptable for both parties. The final state is: 1) The number of columns is equal to the number of lines. 2) Both are infinite, according to set theory this means omega or aleph_0. But you believe: omega or aleph_0 is the maximum of the set of lines and the supremum of the set of columns, which, however, is not taken by any column. You believe the different meaning is not important. I believe that a taken maximum and a not taken supremum are so different (although the difference is very tiny) that set theory, necessarily assuming this, is wrong. The difference between maximum und supremum is, by the way, the reason why I insist that an actually infinite set of natural numbers must contain a non-natural number. > > D can be projected or mapped into a line. Then it is a line longer than > > any line. > > > > D cannot be projected or mapped into any line. You can map d_kk --> d_k. > > You wish to introduce a line with omega letters? > > That yields what I always say: There is no infinite number of natural > > numbers without an infinite number omega. > > All you are doing is calling the infinite number of natural numbers > omega. > Then you say, there is no omega without omega. I think this is obvious. > You do not want to > merely assume > that omega doesn't exist. You want to show that assuming its existence > leads > to a contradiction. If we do not intermingle supremum and maximum, it does so in fact. > > > > You don't get a contradiction if you assume that a supremum is taken > > which, by definition, is not taken? > > > > What definition says that a supremum is not taken? The definition of natural numbers as finite numbers: A n : n < omega. Regards, WM
From: mueckenh on 3 Nov 2006 07:20 David Marcus schrieb: > > > Let's make it simple. I'll give a statement and you say whether you > > > think it is provable in ZFC. Is > > > > > > The set of natural numbers is infinite > > > > > > provable in ZFC? Please answer "Yes" or "No". > > I see you didn't answer my question. Did you really expect an answer? Yes, it is provable in ZFC. I don't know anybody who would deny this. > > > > > > As for the current definition of omega, Kunen's book is a good > > > reference. According to Kunen, omega is not a natural number. > > > > That is out of any question. > > Sorry, but I don't understand. Please rephrase. It is somewhat tedious to correspond with you. Every natural number is a finite number by definition. > > > > I'm > > > guessing that by "whole number" you mean natural number, but I really > > > don't know, since you seem to have your own language for everything and > > > you never give definitions for any of the words that you use. > > > > "Whole number" is Cantor's name for his creation. > > > > My question is : Do you maintain omega > n for all n e N? > > Before I can answer the question, I need to know what you mean by the > words/terms. So, please define "omega", "N", and ">". Also, by > "maintain" do you mean that ZFC proves it? Do you keep on thinking that omega > n for all N, applying all words/terms as Kunen would understand them. Regards, WM
From: mueckenh on 3 Nov 2006 07:32 Dik T. Winter schrieb: > > omega is the supremum, not the maximum. It does not contribute a > > diagonal digit. > > And it does not contribute a line. But the number of lines is omega. > And the number of digits is omega. I think we have obtained agreement, that we disagree in this point and that further discussion will not lead to a result acceptable for both parties. The final state is: 1) The number of columns is equal to the number of lines. 2) Both are infinite, according to set theory this means omega or aleph_0. But you believe: omega or aleph_0 is the maximum of the set of lines and the supremum of the set of columns, which, however, is not taken by any column. You believe the different meaning is not important. I believe that a taken maximum and a not taken supremum are so different (although the difference is very tiny) that set theory, necessarily assuming this, is wrong. The difference between maximum und supremum is, by the way, the reason why I insist that an actually infinite set of natural numbers must contain a non-natural number. > > An infinite diagonal requires not only an infinite length but also an > > infinite width of the matrix. Therefore your absurd infinite number of > > finite lines does not help you. Here we have the same facts as in our > > old problem > > 0.1 > > 0.11 > > 0.111 > > ... > > > > you remember? Without an infinite number in the list there is no > > infinite diagonal defined. > > I state (you know that) that the diagonal: 0.111... Yes, again mixing up maximum and supremum. > > > > > The > > > > diaogonal cannot be roader than the list. The length of the diagonal is > > > > the minimum of width and length. This knowledge is prior to your > > > > axioms. > > > > > > Width and length are equal. > > > > Fine. But the width is finite by definition. (We do not put finite > > segments together, but we have only finite seqments.) > > Nope. The width is unlimited, as is the length. And so is not finite. But the length is realized as a whole number by a set of elements (lines) while the width is not realized by a set of columns. > > > > > A matrix with width A and length B has a diagonal which has min(A,B) > > > > elements. If your axiom contradicts this, then the axiom contradicts > > > > mathematics and should be abolished. > > > > > > That is not contradicted. Width and length are equal. > > > > This amounts to say that there are infinite natural numbers or that the > > diagonal is longer than any line. > > Wrong. If you map the diagonal on a line by d_kk -> d_k, then the line (d_k) is longer than any line (a_mk) for any m e N. Therefore the diagonal has more elements than any line (if omega > m for all m e N). > > > The maximum of a set of finite numbers which has no maximum is simply > > not present. > > Right. > > > It is *not* an infinite number which is larger than any > > finite number, because a maximum must belong to the set. > > Right. > > > And a supremum > > not belonging to the elements of the set does not yield a diagonal > > digit. > > And again right. And still your conclusion is wrong. Obviously the diagonal > has no finite length, because if it had it would have the same length as > one of the numbers, and then the next number would be longer, which is > a contradiction. So the diagonal is infinite in length. And nevertheless its length is assumed to be described by a (non-natural) number. (One cannot compare numbers with non-numbers.) > > > > > And if you can conclude that in this context every straight line > > > > crosses itself 17 times, then you will also take that as a fact? > > > > > > You, if that follows from some axiom, it would really be possible, unless > > > the added axiom leads to an inconsistencey. But I think there might be > > > surfaces where that is even valid. > > > > Let us stick to Euclidean geometry. But that is unimportant. I see it > > is impossible to convince you of the existence of reality. > > What is the "existence of reality" in this context? That what exists, that was really is, contrary to a straight line which crosses itself. Regards, WM
From: mueckenh on 3 Nov 2006 07:46 Dik T. Winter schrieb: > > > No, something quite different was argued there. Namely that the limit > > > *also* is the number of edges in the infinite tree (or somesuch) > > > requires transfinite induction. > > > > But is does not. > > Indeed, it does not. But I thought you were maintaining that it would be? Good heavens! I should require transfinite induction? > You need transfinite induction to show that "the number of edges in the > infinite tree" is equal to the limit of "the number of edges in the > finite tree". No. The number of edges in a finite tree is without interest. We consider only the infinite tree. The set of edges there can be enumerated like the set of rational numbers, for instance. If a mapping N --> Q is defined for every element of Q, then Q is countable. The mapping N --> {edges} has been established such hat every edge knows its number. Regards, WM
From: mueckenh on 3 Nov 2006 08:04
Dik T. Winter schrieb: > In article <1162470378.676172.129570(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> muecke= > > nh(a)rz.fh-augsburg.de writes: > ... > > > > > > > And in mathematics 1/oo is *not* defined. > > > > > > > > > > > > Not in mathematics. But in a theory which assumes omega to be > > > > > > a whole number. > > > > > > > > > > Which theory? > > > > > > > > Set theory. > > > > Cantor invented omega and defined omega as a whole number. > > > > Who changed this standard meaning? > > > > Why do you think this meaning was changed? > > > > When do you think the contrary meaning became standard? > > > > What is the contrary meaning? > > > > Do you agree that A n: n < omega is incorrect? > > > > If not, why do you complain about on-standard meaning on Cantor's > > > > definition? > > > > > > A nice rant. > > > > Why don't you answer? > > Because it is unrelated with the question where 1/oo was defined. > But to answer, I do not know who and when the definition of ordinal > number was introduced. I know *why* it might have been introduced: > to remove possible confusion. I have no idea what you mean with > "contrary meaning". As your statement A n: n < omega does not > indicate where n is coming from, I can not state whether it is > correct or not. But if the natural numbers are meant, than it is > certainly correct (if we assume the natural comparison between ordinal > numbers and natural numbers). n e N was implied. If you can compare two entities in some dimension like length or weight , then they are of the same sort with respect to this dimension. > > > > > Where did Cantor define 1/oo? A quote might be > > > appropriate. > > > > see below > ... > > > Why? As long as 1/omega is not defined you can not talk about it. You > > > simply assume that 1/omega is a number. But that is not the case, it > > > is not defined. > > > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > > N. > > First, how do you *define* 1/omega? You know how arithmetic normally is > defined? But of course, once you define it, some properties of arithmetic > are lost. What is omega * (1/omega)? This property is lost. > > > > > In particular the limit of all segments of N is not defined. But set > > > > theory does it. > > > > > > No. And if so, show me where. > > > > Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu > > denken, welcher die Zahlen n zustreben, wenn darunter nichts anderes > > verstanden wird, als daß omega die erste ganze Zahl sein soll, welche > > auf alle Zahlen n folgt, d. h. größer zu nennen ist als jede der > > Zahlen n. (p. 195) > > > > This definition has been conserved up to our days: Limesordinalzahl or > > limit ordinal number. > > Yes. But "limit" has not really been defined. Grenze ist immer an sich etwas festes, unveränderliches, daher kann von den beiden Unendlichkeitsbegriffen nur das Transfinitum als seiend und unter Umständen und in gewissem Sinne auch als feste Grenze gedacht werden. > > > > > Everything is a set (in ZF). Numbers are sets too. > > > > > > In the Von Neumann model of ZF, I would think. > > > > In ZF. > > > > Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel > > Dekker Inc., New York, 1984, 2nd edition, p. 2: "So the only objects > > with which we are concerned from now on are sets." > > Again, depends on the model. Of course. But the usual set theory without yet specifying any models starts from this point. Everything is a set is true in theories like ZF, with restricted comprehension. Unrestricted comprehension requires a universal set. In particular, the powerclass of the universal set is not a set, in that case. ZF doesn't have a universal set. In NBG, not everything is a set (nor is "everything" a set in anything with a universal set). > But even if we allow that, it can not > be applied to the ball and vase problem. In that case there is a > limit involved (in your models of that problem). And to use those > limits (as you do) you have > (1) to define the limit of a particular sequence of sets > (2) to show that the cardinality of that limit is equal to the > limit of the cardinalities. Wrong. The only thing to be shown is that omega is the limit of the natural numbers. This is the fundamental assumption of set theory. Multiplication by 9 is negligible. Regards, WM |