Cantor Confusion
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From: Dik T. Winter on 6 Nov 2006 20:31 In article <1162820871.886446.129490(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1162557178.206693.187650(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > 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. > > > > There is not a taken maximum. > > Oh no? The lengths of the columns of the list > > 0.1 > 0.11 > 0.111 > ... > > are not omega? I was under the impression that there are aleph_0 lines > with less than aleph_0 columns. There are alpeh_0 lines and aleph_0 columns. But neither is a maximum. Both are supprema. There is no aleph_0-th line as there is no aleph_0-th column. So, where is there a maximum involved? > > > 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. > > > > Yes, you keep insisting that, but as there is no maximum, that insistance > > is futile. > > The number 0.111... or the diagonal has alep_0 1's as has the n-th > column but there i sno line having aleph_0 1's. Yes. What does that prove? *Not* that the infinite set of natural numbers contains a non-natural number. Because that is obviously nonsense. > > But let's take the actually infinite set of natural numbers. > > Suppose it contains your non-natural number. Now remove that non-natural > > number from that set. Isn't the set still actually infinite? > > That shows only the self-contradiction of the notion actual infinity. No self-contradiction at all. What *is* the self contradiction? Your remark only shows that you do not understand what the actually infinite set of natural numbers is. > > What you are confused about is that the cardinality (and ordinality) of > > the set of natural number is indeed actually infinite; but the *contents* > > are only potentially infinite. > > You see not difference between the infinite man 1's of 0.111... and the > finite many 1's in every line? I see a difference, but I have no problem with that difference. > > > > > 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. > > > > No, not mixing at all. Supremum only. > > Then 0.111... is not different from the finite sequences of 1's? How do you conclude that? Given finite initial segments of the triangle we get a length, a width and a length of the diagonal that are all equal. When we look at the complete triangle (which exists by the axiom of infinity) all three become the supremum of the finite quantities, and so still are equal. No maximum involved. > > > 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. > > > > You are confused between two concepts. The length is not realised in the > > sense you think it is. > > The length every column is infinite and larger than any n while no line > is. Yes. What is the problem with that? The length is realised as omega as the supremum of all finite lengths, so is the width realised as omega as the supremum of all finite widths. Where is the difference? > > > 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). > > > > Right. The diagonal has more elements than any line. > > But not more than any column. Why is there a difference? Because you let the triangle grow without bound. The length of any column is the supremum of the lengths in the finite triangles. The length of the diagonal is the supremum of the lengths of the diagonals in the finite triangles. But there is no actual line at position omega, so there is also no actual column at position omega. If there was such an actual line there would also be such an actual column. > > That is obviously > > true by its definition. > > The sentence "I am greater than everybody including myself" cannot be > true by definition. Depends on the definition. > > Yes? What is the problem with that? The length of each and every line > > is described by a natural number, and each and every natural number > > describes a line. So if we want to describe the length of the diagonal, > > it can not be a natural number. That length is called omega (if we want > > to also maintain the ordering information) or aleph-0. > > But you don't see a difference between a diagonal having aleph_0 1's > and the lines which it is constructed from which have less 1's? No. Just as I do not see a difference between a triangle having aleph_0 lines which is constructed from triangles that have fewer lines. The principle is simply the same. > > > > What is the "existence of reality" in this context? > > > > > > That what exists, that was really is, contrary to a straight line which > > > crosses itself. > > > > What do you mean with "that what exists, that what really is"? > > If you can't understand it then nobody can teach you. If you cannot define it, nobody will understand it. -- 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 6 Nov 2006 20:46 In article <1162821418.935074.235580(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > Sorry, I misread. You insist that the limit *also* is the number of edges > > in the infinite tree. To prove that you need transfinite induction. > > Then you need transfinite induction to prove that the number of > rationals in the case of an infinite set Q is countable. How wrong you are. In the case of the edges you show that for finite trees of order n: #edges(n) = f(n) with some function n. From that you conclude: #edges(oo) = f(oo) which requires transfinite induction. To prove that Q is countable you do not need infinity at all. See the construction of a bijection I gave from N to Q+, using simplified continued fractions. This shows that every finite n maps to a finite q>0 and the reverse. So we have a function g: N -> Q+ that has an inverse. And that is all that is needed to show countability. Where does the tranfinite induction come in? > > > 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. > > > > You are wrong. > > Please try to understand the concept of countability. Only in the case > of infinite sets it is of interest. Cantor defined it without (and > before he was knowing anything about) transfinite induction. Indeed. To prove countability you do not need transfinite induction. But my remark "you are wrong" was to your statement: "The mapping N -> {edges} has been established". See J. H. Conway, On Numbers and Games, for a clear exposition about the difference between the union of all finite trees and the infinite tree. -- 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 6 Nov 2006 21:11 In article <1162825586.068932.25310(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > 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. > > > > Perhaps. But we do not talk here about length or weight. We are talking > > mathematics. > > How else lengths and weight could be compared if not by mathematics? Weights by scales. Lengths by putting them alongside. No mathematics involved. But at least weight is not a mathematical entity. Length is, but not in the way you see it. > > You can compare two entities of a different kind if, and > > only if, a comparison function is defined. But as the natural numbers > > can be embedded isomorphically in the ordinal numbers, there is a > > natural comparison function. > > Why then do you oppose to state omega > n e N and to draw conclusions > from that. I did not oppose it. I only stated that it was true if you did use the natural comparison function. > > Still, how do you *define* 1/omega? You just state 1/omega = 0? Oh, > > well. > > I state omega > 2 which implies 1/omega < 2. Then I take 3 instead of 2. > And so on. That is not a definition. The first thing to do even that you have to posit that 1/omega is something. Next you have to posit that 1/omega has some properties from ordered fields, and you have to state what properties it has. Finally you can make the above statement. And may still to fail to find that 1/omega = 0. In Conways numbers, 1/omega is certainly different from 0, while omega has the same role as omega from Cantorian arithmetic. (They embed the ordinals but with a slightly different arithmetic.) > > Perhaps. That makes you indeed lose a few arithmetical properties, so > > you can not use them in general, but must use them with extreme care. > > The property above is not lost. Which property? > > > > Yes. But "limit" has not really been defined. > > > > > > Grenze ist immer an sich etwas festes, unver?nderliches, daher kann > > > von den beiden Unendlichkeitsbegriffen nur das Transfinitum als seiend > > > und unter Umst?nden und in gewissem Sinne auch als feste Grenze > > > gedacht werden. > > > > Still, I see *no* definition of "limit". > > Small wonder. Indeed, as there is no such definition. Have a bit of education in Topology and you will find that not every border is also a limit. Limits are outside the context of set theory. They belong to topology. > > > Of course. But the usual set theory without yet specifying any models > > > starts from this point. > > > > Not when I followed the courses on set theory. > > Why didn't you? I did. Why do you think I did not? Did *you* follow a course on set theory when you were a student at some university? At least at the university where I did study they were required courses for both mathematics and physics. -- 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 8 Nov 2006 05:02 William Hughes schrieb: > > Oh no? The lengths of the columns of the list > > > > 0.1 > > 0.11 > > 0.111 > > ... > > > > are not omega? I was under the impression that there are aleph_0 lines > > with less than aleph_0 columns. > > Be careful with your quantifiers. The fact that every line has less > than aleph_0 columns does not mean there are less than aleph_0 > columns, it just means that no line has all columns. It just means that not all columns are there. There is no actual infinity. Interesting is that all columns have aleph_0 lines. So all lines are there. You should at least be able to see the difference between the actuality of the aleph_0 lines which are present, which are realized, and the aleph_0 columns which are not realized and therefore, are not present. > > There are aleph_0 columns. > Each line has less than aleph_0 columns. > No line has all columns > Every column is in some line. This is potential infinity. No line has aleph_0 columns. There is another kind of infinity: Every column has aleph_0 lines? Got it? > > > > > You see not difference between the infinite man 1's of 0.111... and the > > finite many 1's in every line? > > I see a difference. However, I do not see a contradiction. 111... > is not one of the lines and has a property that none of the lines > share. 111... can be mapped on a line, say the first line. Then the first line has aleph_0 1's. As no line has so much 1's, now the first line is longer than all other lines. > > > > > > > > > > 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. > > > > > > No, not mixing at all. Supremum only. > > > > Then 0.111... is not different from the finite sequences of 1's? > > Yes is it different. So what? The supremem of A may not be a member > of A. The supremum of A can be different from every element of A. But why is it not different from the columns? > Again this is true but there is no contradiction. The number of > columns > is not the maximum of the column lengths but the supremum. This supremum is a thing which is not realized by the number of columns but which is realized by the numer of lines (if the notion of finished infinity is correct). What ma ybe the reason for this difference? > Because, by construction, every column is infinite and no line is. > Why do you insist this simple fact is a contradiction? Because in natural numbers we have n = n, e.i., the n-th number has size n. If you transpose the matrix, then nothing can change. But in our matrix, we have a change. This is not possible for natural numbers. > > But you don't see a difference between a diagonal having aleph_0 1's > > and the lines which it is constructed from which have less 1's? > > No. Because the diagonal is the supremum, and the supremum of a finite > set does not have to be finite. Maybe. But a diagonal cannot exist in a domain where no lines reach. So, if omega is something realized, then there is a contradiction. Regards, WM
From: mueckenh on 8 Nov 2006 05:19
William Hughes schrieb: > The set of lengths of columns, C, consists of the single element > omega. This is a maximum taken by every length of a column. > The set of widths of lines, W, is the set of all natural > numbers. > W has more elements than C, but every element of W is > smaller than every element of C. Correct. > > The supremum of W is omega. The width of the > matrix is the supremum of the set of widths of the lines, i.e. > omega. Thus the width of the matrix is equal to the height of the > matrix. But this supremum is not taken. The diagonal connects width and length by a bijection. The element d_nn of the diagonal maps the n-th line on the first n elements of the first column. As long as n is a natural number, this is no problem. Only for aleph_0 the diagonal has to map a not existing maximum on an existing maximum. This is a hard task. (But it occurs only if aleph_0 is assumed to exist.) Regards, WM |