Cantor Confusion
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From: mueckenh on 21 Jan 2007 11:14 Virgil schrieb: > In article <1169332384.876640.319580(a)a75g2000cwd.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > > > > > Whether a _meaning_ of > > > > > > T(1) U T(2) U T(3) U ... (inf-u) > > > > > > exists does _not_ depend on such question. It depends on a definition > > > you have to provide. So please answer the question, what (inf-u) means > > > and prove that it exists. > > > > > > > This is connected > > > > > > Whether that is "connected" is irrelevant to a possible definition of > > > (inf-u). > > > > That is not a true statement. Nevertheless: > > > > Definition of the infinite union of trees by induction: If the finite > > tree with n levels exists, the finite tree with n+1 levels exist. The > > tree with 1 level exists. > > Therefore, by induction, all finite n-lever trees for all n in N exist. > But that is all that standard induction allows one to conclude. It is enough. Or should there one level be missing? Please specify which remains to be included. > > > > Proof of existence of the union tree by proofs of A, B, and C: > > A) Proof of the existence of one infinite path by induction over the > > indexes > > {1} U {1, 2} U {1, 2, 3} U... U {1, 2, 3, ..., n}... U ... > > is the same as the union of the ends of the initial segments > > {1} U { 2} U {3} U... U {n}... U .. > > which is N which exists. > > B) Proof of the existence of all infinite paths: See proof of the > > existence of all real numbers in [0, 1]. > > C) Proof of the existence of all paths in the tree being infinite:: All > > paths of a tree have same length by definition. > > First of all, the standard definition of finite binary trees does not > require that all paths in such a tree be of the same length. But the definition given by me requires it. So it holds for both types of my trees, the cut tree as the weeping willow. > It is only > in our special examples here that we are making such a requirement, but > it is not "by definition". > > Secondly, there are lots of different infinite binary trees in which > every path is of infinite length. The one of major interest is that one > having one path corresponding to every possible binary sequence, which > we have been calling the complete binary tree. It has uncountably many > paths. > > But there are also infinite binary trees containing the same set of > nodes and the same set of edges but having only countably many paths. You assert so, yes, we know. But how can a tree prohibit a possible path to exist? All the nodes and edges guiding the path are there. What hinders the path to exist? Is it the spirit required to understand set theory? > > For example, the infinite binary tree having only "eventually constant" > paths (all branchings are 0's or all 1's from some node on) has exactly > the same set of nodes and exactly the same set of edges as the complete > binary tree. > > So the range of such infinite binary trees includes both ones with only > countably many paths and ones with uncountably many paths. The first type of trees is certainly existing. But how can we conclude that the second type does exist? Isn't it the conclusion that the representation of a number with N digits is the representation of a real number? Regards, WM
From: mueckenh on 21 Jan 2007 11:26 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > William Hughes schrieb: > [...] > > I say: The union of indexes of paths {1} U {1,2} U {1,2,3} U ... is an > > infinite index union {1,2,3,...} > > "{1} U {1, 2} U ..." means U {{1, ..., n} | n e N }. This set-union > exists due to the axiom of union (in ZFC). It is valid that > > s = sup {{1, ..., n} | n e N} = {1, 2, ...} = N > > Note that N is not a member of s, N is not the maximum of > {{1, ..., n} | n e N}. N is the union of all of its elements. N is the union of al of its initial segments. > > > corresponding to n infinite path. > > Where is your union of all finite trees? You have not yet shown: It is interesting: William knows that T1 and T2 exist, but T1 is different. Virgil is also accepting T1 but, like William, sees only finite trees therein. Dik knows that T1 contains infinite trees, but doubts that its paths are the union of the paths of the finite trees. You doubt the existence of T1 (seeing that otherwise set theory is contradiced?). Couldn't you get to a consensus? It would spare me a lot of work. > > ,----[ <45af6ca0$0$97262$892e7fe2(a)authen.yellow.readfreenews.net> ] > | 7. Now let V* denote the set of all finite trees { T(i) | i e N }. > | > U V is only defined for V having card(V) e N. Since V* does not > | meet this requirement we (you?) have to define what > | > | U V* > | > | shall mean. The obvious definition > | > | "U V* = T(max(D(V*))" > | > | fails due to the reason that max(D(V*)) = max(omega) is not defined. Max n,m is not defined either, nevertheless the union of {1,...,n} and {1,..., m} and the infinite union of all segments are defined. > | > | The questions is: How do you define U V_omega? There is no question open. The union of the finite initial segments of N is N. This can be applied to the path 0.111... This can be carried over to all paths, except in case that not all paths exist (but only 0.111... and other periodic paths, or not even they). Regards, WM
From: Franziska Neugebauer on 21 Jan 2007 11:42 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > William Hughes schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > William Hughes schrieb: >> [...] >> Where is your union of all finite trees? You have not yet shown: > > It is interesting: > William knows that T1 and T2 exist, but T1 is different. > Virgil is also accepting T1 but, like William, sees only finite trees > therein. > Dik knows that T1 contains infinite trees, but doubts that its paths > are the union of the paths of the finite trees. > You doubt the existence of T1 (seeing that otherwise set theory is > contradiced?). I want a _definition_ of the notion T(1) U T(2) ... (the infinite union of all finite trees) you have not yet provided. > Couldn't you get to a consensus? It would spare me a lot of work. >> >> ,----[ <45af6ca0$0$97262$892e7fe2(a)authen.yellow.readfreenews.net> ] >> | 7. Now let V* denote the set of all finite trees { T(i) | i e N }. >> | >> U V is only defined for V having card(V) e N. Since V* does not >> | meet this requirement we (you?) have to define what >> | >> | U V* >> | >> | shall mean. The obvious definition >> | >> | "U V* = T(max(D(V*))" >> | >> | fails due to the reason that max(D(V*)) = max(omega) is not >> | defined. > > Max n,m is not defined either, Aha ,----[ WM in <1169111380.377993.67320(a)l53g2000cwa.googlegroups.com> ] | The union of two finite trees T(m) and T(n) with m and n levels, | respectively, where m < n, is the tree with n levels. `---- So you mean m < n is not defined? Then it makes no sense at all to write about trees? 3b. Definition of max(a, b): Let a /= b be members of some ordered set (S, <). Then max(a, b) is a iff b < a and b iff a < b. > nevertheless the union of {1,...,n} and > {1,..., m} and the infinite union of all segments are defined. We are writing about trees. >> | >> | The questions is: How do you define U V_omega? > > There is no question open. There is still an open question: How do *you* _define_ The infinite union of all finite trees? > The union of the finite initial segments of N is N. We are writing about trees. F. N. -- xyz
From: David Marcus on 21 Jan 2007 12:52 mueckenh(a)rz.fh-augsburg.de wrote: > It is interesting: > William knows that T1 and T2 exist, but T1 is different. > Virgil is also accepting T1 but, like William, sees only finite trees > therein. > Dik knows that T1 contains infinite trees, but doubts that its paths > are the union of the paths of the finite trees. > You doubt the existence of T1 (seeing that otherwise set theory is > contradiced?). > > Couldn't you get to a consensus? It would spare me a lot of work. The problem is that people use different definitions. It would help if each person would include their definitions of important objects (e.g., T1) in each post. -- David Marcus
From: Carsten Schultz on 21 Jan 2007 12:55
mueckenh(a)rz.fh-augsburg.de schrieb: > It is interesting: > William knows that T1 and T2 exist, but T1 is different. > Virgil is also accepting T1 but, like William, sees only finite trees > therein. > Dik knows that T1 contains infinite trees, but doubts that its paths > are the union of the paths of the finite trees. > You doubt the existence of T1 (seeing that otherwise set theory is > contradiced?). > > Couldn't you get to a consensus? It would spare me a lot of work. Since your writings are so confused and you never give exact definitions or arguments, people have to guess what you mean. Sometimes they guess differently. It would save everybody a lot of time if you would be precise in your statements. However, you do not seem to be capable of this. Also, the errors in your arguments would be even more obvious, maybe even to you, if your arguments were stated precisely. [...] > The union of the finite initial segments of N is N. > This can be applied to the path 0.111... > This can be carried over to all paths, except in case that not all > paths exist (but only 0.111... and other periodic paths, or not even > they). Err, yeah. Nice example. -- Carsten Schultz (2:38, 33:47) http://carsten.codimi.de/ PGP/GPG key on the pgp.net key servers, fingerprint on my home page. |