Cantor Confusion
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From: Dik T. Winter on 13 Nov 2006 20:22 In article <1163431235.417059.113430(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > Well, my only advise is, read it. > > > > > > If he says so, then it wil not be a good idea to waste my time with it. > > > > Do you really think the node 1/3 is finitely far from the root in the tree? > > Dik, are you joking? There is no node yielding any number like 1/3. > Every node represents one bit of the binary representation of many real > numbers. You switch so many times from what represents what, that I lost track. But with this notation: there is no node in the tree where 1/3 is completed. If there were such a node, it would be infinitely far away. But, whatever. You can just as well state that each node carries also the decimals above it, the distinction is extremely small. > The first two levels of the tree contain the following initial > segments: > 0.00... > 0.01... > 0.10... > 0.11... Note that at every level n of your tree the numbers represented are all of the form a/2^n, with 0 <= a < 2^n. So neither 1/3 not 1/5 are in any of the finite subtrees. > > Your node 1/3 is infinitely far from the root because there is no finite > > (natural) number that can state the distance. > > There is no node 1/3. There is the number 1/3 consisting of infinitely > many nodes 0.010101... None of them has infinite distance from the > root. But that one is not in the tree. If (as I state above) we consider that each node also carries the decimals above it (which is equivalent to your statement), each rational number in [0, 1) with a power of 2 in the denominator is in the completed tree, but no other numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: William Hughes on 13 Nov 2006 21:04 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Cardinal numbers like one, two, ... and ordinal numbers like first, > > > second, ... are closely connected. So every iinitial segment of natural > > > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > > > Therefore no initial segment of natural numbers can be counted by an > > > unnatural number like omega. > > > > Only if we say that {1,2,3,...} is not an initial segment of natural > > numbers. > > It is an initial segment of natural numbers because it contains only > such numbers in well-ordering. O.K. We use a definition such that N={1,2,3,...} (with the ... representing only natural numbers) is an intial segment. Now N has no maximum element. If a natural number n counts an intial segment of natural numbers B, then n is the maximum elment of B. Since there is no maximum element of N, there is no natural number that can count n. > > > > No set of the form {1,2,3..,n} can be counted by an unnatural > > number like omega. The set {1,2,3,...} is not a set of the form > > {1,2,3,...,n} > > If only natural numbers are within, that it is of that form. Yes N={1,2,3,...} contains only natural numberss. However, unline a set of natrual numbers of the form {1,2,3..,n}, N does not have a largest or last number. > we may not know the last one, we can be sure that it is a natural. No, the reason we do not know the last one is that there is no last one. > > > > > This leads to the problem |{1,2,3,...}| = > > > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > > > a or a + 1, corresponding to the kind of a. > > > > Yes. This is true. It is not however a problem. > > It shows that by setting omega we loose a number in between. It should > disturb a mathematician It just shows that sets of ordinals behave differently than sets of natural numbers. This is exaclty what a mathematician expects. .. > > > > [Sometimes this is used to argue that starting at 0 is more elegent. > > That is nonsense. Piffle. >The first natural number is that one which is called > the "first" and not the zeroth. And the first ordinal is the one that is called the "zeroth" not the first. Which is not to say that counting ordinals is more correct than counting natural numbers and then extending this count to the ordinals. It is however more elegent. >The set {1} has 1 element, the set > {1,2,} has 2 elements and so on. > Right up through every natural number n, with the set initial segment ending at n, I_n {1,2,3,...,n} having n elements. In this way we use all the natural numbers. But wait. There is one initial segment we haven't counted yet. No set I_n contains all the natural numbers. So lets count the set of all natural numbers {1,2,3,...} There are no natural number left. So we stop using natural numbers and use ordinals (and to nobody's surprise a few things change). The set {12,3,...} has omega elements, and the set {1,2,3,...,omega} has omega + 1 elements, and the set {1,2,3,...,omega,omege+1} has omega + 2 elements etc. Now for a set of the form {1,2,3, ... , a} the ordinal of the set is a+1. This is because sets of ordinals do not act quite like sets of natural numbers. - William Hughes
From: William Hughes on 13 Nov 2006 21:20 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Well, since I already agree there is no infinite column this does > > not change anything. > > If you add only one element to each column, you get the order type > omega + 1 for the length of he matrix. > If you add only one element to each line, the order type of the width > of the matrix remains omega. > Correct. No column has an infiite term. Each column has as infinite number of terms. No line has an infinite term No line has an infinite number of term The supremum of the number of line indexes is the same as the supremum of the number of column indexes. However, the fact that the supremums are the same does not mean that the sets are the same. If two sets A and B are not the same, adding one element to A may not give the same result as adding one element to B. > > > > > There > > > > are an infinite number of lines. Each line is finite. > > > > > > There is an infinite number of initial segments of columns. Each one is > > > finite > > > > Which is just what I said (changing names from "line" to "initial > > segments > > of columns" changes nothing). > > Wrong. By lines I understand the horizontal rows of a matrix. By > columns I understand the vertical rows. And by "initial segments of columns" you mean a vertical initial segment. Yes, there are an infinite number of initial sements of colums. Each one is finite. The length of a column is the supremum of the lengths of these initial segments. The length of a column is not the maximum line index, because there is no maximum line index. - William Hughes
From: William Hughes on 13 Nov 2006 21:34 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > William Hughes wrote: > > > > > > > > > a: Any infinite set of numbers must contain an infinite number > > > > > > > > > > b: It is possible to have an infinite set of numbers that does > > > > > not contain an infinite number. > > > > > > > > WM, please tell us if you agree or disagree with the statements a and b > > > > above. > > > > The potentially infinite set N does not contain an infinite number. In > > > this case (b) is correct. > > > > > > An actually infinite set is a set which has a cardinal number like > > > omega. > > > > You are confusing cardinals and ordinals. For finite sets they > > are the same. For infinite sets they are different. You > > want ordinals. > > The cardinal number aleph_0 is the same as the cardinal number omega. > Already Cantor in his later years use omega as a cardinal. Names do not matter. Let kumquat be the first infinite ordinal. kumqual + 1 is not equal to kumqual. Let breadfruit be the first infinite cardinal. breadfuit +1 equals breadfruit. You want kumquat behaviour so you want ordinals. However, the ordinals up to omega behave almost exactly like the cardinals up to aleph_0, so if we don't go any further we don't have to worry about the difference. > > > > This cardinal (actually ordinal) number is not an element of the set. > > May be so with some sets. A set of natural numbers includes its > cardinal number. No. (not even if we change this to initial segment of natural numbers which is what you mean). An initial segment of natural numbers that has a maximum contains its cardinal. There is one initial segment of natural numbers, {1,2,3,...} that does not have a maximum. {1,2,3,...} does not include its cardinal number. > > > > > In set theory every set is actually infinite. So N is actually > > > infinite. If its cardinal number omega is claimed to count the members > > > of N, then there is a contradiction, because omega counts all n in N > > > but does not belong to N while the initial segments of N are counted by > > > natural numbers. This statement remains true as far as the natural > > > numbers reach, i.e., for all natural numbers , i.e., for N. > > > > BZZZ > > > > The fact that something is true for all sets of the form > > {1,2,3,...n} where n is a finite natural number, > > does not mean that it is true for N. > > Oh yes, exactly that it means, because N consists of nothing else than > natural numbers. There are no ghosts in mathematics. > > > > For example: > > For every set B of the form {1,2,3,...,n} there > > exists a finite natural number m such that B={1,2,3,...,m}. > > However > > there is no finite natural number m such that N={1,2,3,...,m} > > [This is true even if we assume that N consists of exaclty those > > natural numbers that will be named during the lifetime of the > > universe]. > > As long as the "..." denote nothing but natural numbers, your statement > is obviously false, as induction proves. No. Induction can only prove things about initial segements of the natural numbers that have a maximum. Induction cannot prove anything about an initial segment of natural numbers that does not have a maximum. One such initial segment N={1,2,3,...} (where the "..." denotes only natural numbers) exists - William Hughes
From: William Hughes on 13 Nov 2006 21:45
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Every column has a fixed number omega of terms. You can add 1 further > > > term and you will get omega + 1 terms in every column. Every line has > > > less than omega terms. If you add 1 term to every line, the number of > > > terms in every line remains less than omega. What about the diagonal? > > > It has to satisfy both conditions, which is impossible. > > > > No. Adding one element to each line does not change > > the supremum of the lengths of the lines, so it does > > not change the length of the diagonal. > > The supremum is larger than any line, so it must be larger han the > diagonal which is confined by the lines. No. The diagonal is "confined by the lines" only in the sense that the length of the diagonal is the supremum of the lenghts of the lines. Thus if there is no line with maximum length, the diagonal will be larger than any line. > > > > The number of terms in each line, n, is less than omega. > > The number of columns is the supremum of the number of terms > > in each line. The number of columns is omega. > > If there is no line with omega terms, what does the supremum of the > columns consists of? > Nothing. The supremum of the columns does not exist. However, the supremum of the number of terms in each line exists. This is omega. It is also the number of columns. (Note the fact that there are an infinite number of columns does not mean that there is an column with infinite index). > > The number of lines is omega. > > The number of columns is equal to the number of lines. > > You can repeat your prayer as often as you like. It will not be heard. Clearly. But if you define the number of lines as the supremum of the number of terms in each column, and the number of columns as the supremum of the number of terms in each line, it is true. > > > > > > Now add one term to each line. > > > > The number of terms in each line, n+1, is less than omega. > > The number of columns is the supremum of the number of terms > > in each line. The number of columns is omega. > > The number of lines is omega. > > The number of columns is equal to the number of lines. > > You can repeat your prayer as often as you like. It will not be heard. > Clearly. But if you define the number of lines as the supremum of the number of terms in each column, and the number of columns as the supremum of the number of terms in each line, it is true. (Yes I can cut and paste too.) - William Hughes |