Cantor Confusion
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From: mueckenh on 12 Oct 2006 06:52 Dik T. Winter schrieb: > In article <1160578706.221013.145300(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > So the definition I gave for a limit of a sequence of sets you agree > > > with? Or not? I am seriously confused. With the definition I gave, > > > lim{n = 1 .. oo} {n + 1, ..., 10n} = {}. > > > > Sorry, I don't understand your definition. > > What part of the definition do you not understand? I will repeat it here: > > What *might* be a sensible definition of a limit for a sequence of sets of > > naturals is, that (given each A_n is a set of naturals), the limit > > lim{n = 1 ... oo} A_n = A > > exists if and only if for every p in N, there is an n0, such that either > > (1) p in A_n for n > n0 > > or > > (2) p !in A_n for n > n0. > > In the first case p is in A, in the second case p !in A. > Pray, read the complete definition before you give comments. I do not believe that definition (2) is of any relevance. Cantor uses Lim{n} n = omega witout much ado. omega is simply defined as the limit of the increasing natural numbers. In his first paper he uses even Wallis' symbol oo. What should there require a definition, if all natural did exist? This is what I use and write in modern form: Lim {n-->oo} {1,2,3,..,n} = N. Not this expression is meaningless but the assumption behind it, the complete set N. Regards, WM
From: mueckenh on 12 Oct 2006 07:04 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > I am sure you are able to translate brief notions like "to enter, to > > > > escape" etc. by yourself into terms of increasing or decreasing values > > > > of variables of sets, if this seems necessary to you. Here, without > > > > being in possession of suitable symbols, it would become a bit tedious. > > > > > > Yes, I can translate it myself. However, that would only tell me how I > > > interpret the problem. > > > > Hasn't it become clear by the discussion? > > > > I use two variables for sequences of sets. Further I use a function. I > > use the natural numbers t to denote the index number. The balls are > > simply the natural numbers. I speak of "balls" in order to not > > intermingle these numbers with the index-numbers. > > > > The set of balls having entered the vase may be denoted by X(t). > > So we have the mathematical definition: > > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > > There is a bijection between t and X(t). > > t is a number and X(t) is a set. If t = 1, then your sentence says, > "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. > So, I don't follow. What do you mean, please? That what is written. There is a bijection between the set of all numbers t and the set of all sets X(t). 1 is mapped on X(1), 2 is mapped on X(2), and so on. Is there anythng wrong? > > > The set of balls having left the vase is described by Y(t). So we have > > the mathematical definition: > > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > > There is a bijection between t and Y(t). Here we have the same as above with Y instead of X. > > > > And the cardinal number of the set of balls remaining in the vase is > > Z(t). So we have the mathematical definition: > > Z(t) = 9t with Z(t) > 0 for every t > 0. > > There is a bijection between t and Z(t). Regards, WM
From: William Hughes on 12 Oct 2006 07:06 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > 0.1 > > > > 0.11 > > > > 0.111 > > > > ... > > > > > > > > That is correct. But every element of the natural numbers is finite. > > > Hence every element covers its predecessors. If 0.111... is covered by > > > "the whole list", then it is covered by one element. That, however, is > > > excuded. > > > > > > > Since no one has claimed that '0.111... is covered by "the whole > > list"', I fail > > to see the relevence of a sentence that starts out > > 'If 0.111... is covered by "the whole list"'. > > If every digit position is well defined, then 0.111... is covered "up > to every position" by the list numbers, which are simply the natural > indizes. I claim that covering "up to every" implies covering "every". > Quantifier dyslexia. The fact that for every digit position N, there exists a natural number, M, such that M covers 0.111... to position N does not imply there exist a natural number M such that for every digit position N, M covers 0.111... to position N - William Hughes
From: mueckenh on 12 Oct 2006 07:12 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > You wrote that "A covers B" means that A has at least as many bars as B. > > > > > > Does "S is completely covered by at least one element of the infinite > > > set of finite unary numbers" mean that S is covered by an A that has a > > > finite number of bars? > > > > If S = 0.111... had only finite positions then it was covered by finite > > numbers. All of them are in the list (these are unary representations > > of the natural numbers) > > 1 = 0.1 > > 2 = 0.11 > > 3 = 0.111 > > ... > > If a set of finite numbers of the list cover some number x of the list, > > then always already one list number is sufficient to cover number x. If > > some number y (in unary representation) is not in the list, then it > > cannot be covered by any list number. Then its positions are not well > > defined. This is the case with S. > > Sorry, but I don't know what you mean by "not well defined". I believe > you said (in a previous post) that S is an infinite string of 1's. So, > what do you mean that the positions of S are not well defined? It is impossible to discern the position of some digits. If only 1's are involved, there is not much lost. But in case of numbers with different digits, these numbers are not defined. (I used 0.111... only because it easens the proof that irrational numbers are not well defined.) Regards, WM
From: mueckenh on 12 Oct 2006 07:18
David Marcus schrieb: > Please state an internal contradiction of set theory. Please use the > standard language of set theory/mathematics so that we can understand > what the contradiction is without needing to ask what all the words > mean. Good heavens, there are so many. Where shall I start with? Consider the binary tree which has (no finite paths but only) infinite paths representing the real numbers between 0 and 1. The edges (like a, b, and c below) connect the nodes, i.e., the binary digits. The set of edges is countable, because we can enumerate them 0. /a\ 0 1 /b\c /\ 0 1 0 1 .............. Now we set up a relation between paths and edges. Relate edge a to all paths which begin with 0.0. Relate edge b to all paths which begin with 0.00 and relate edge c to all paths which begin with 0.01. Half of edge a is inherited by all paths which begin with 0.00, the other half of edge a is inherited by all paths which begin with 0.01. Continuing in this manner in infinity, we see that every single infinite path is related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any other path. The set of paths is uncountable, but as we have seen, it contains less elements than the set of edges. Cantor's diagonal argument does not apply in this case, because the tree contains all representations of real numbers of [0, 1], some of them even twice, like 1.000... and 0.111... . Therefore we have a contradiction: Card(R) >> Card(N) || || Card(paths) =< Card(edges) On my homepage you will find the "100 Euro question", stating another contradiction. http://www.fh-augsburg.de/~mueckenh/ Regards, WM |