Cantor Confusion
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From: mueckenh on 16 Nov 2006 14:26 Dik T. Winter schrieb: > In article <1163605613.609012.99490(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1163510733.868272.250410(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > > They have no finite binary representation. But if they can appear in a > > > > list, then they can appear in a tree. > > > > > > Yes, of course they can appear in a tree. But they do not appear in *this* > > > tree. 1/3 can appear in a ternary tree. > > > > So you say that 1/3 does not have a binary representation? > > No, that is specifically *not* what I say. 1/3 does not have a *finite* > binary representation. > > > So you say that pi has no representation at all in a fixed base > > (n-adic or n-ary), and, therefore, cannot appear in Cantor's list? > > No *finite* representation. But "Cantor's list" does allow *infinite* > representations. More specific, his diagonal proof was about lists > where each element consisted of an *infinite* sequence of symbols. Why do you think that my tree does not allow infinite sequences of symbols (nodes)? > > > > > Even of [0, 1] because 0.111... = 1. > > > > If you have this opinion, I will happily agree, but then you must also > > > > apply it to every binary representation of the reals. The tree is > > > > nothing other than such a representation, a special one. > > > > > > And that is wrong. The tree contains only the set of finite binary > > > representations. > > > > i.e., Cantor's original list contains only the set of finite sequences > > of m and w. I agree. > > No "i.e." and wrong. The list was a list of *infinite* sequences. Why do you think that my tree does not allow to represent such sequences with 0 and 1 instead of m a nd w? > > > > > You just stated that there are no infinite strings. A agree. > > > > > > Where did I state that? > > > > Above you said "The tree contains only the set of finite binary > > representations." > > Yup. There is *no* node 1/3. There is no symbol 1/3 in Cantors list. > > > If there are infinite strings elsewhere, then they > > are in my tree too (in form of paths). > > But you had said that it were the nodes that represented the numbers > in the tree, so the paths are irrelevant. I never said so. The nodes represent the bits 0 or 1. > There are infinite paths > in your tree, but they do not contain a node that represents (for > instance) 1/3. So, if the nodes represent numbers (as you have said), Do you have a reference? > 1/3 is not in your tree. You are not clear about what the numbers in > your tree are. Are they the nodes? Are they the paths? Sometimes > you say one thing other times you say something different. So to get > proper understanding. What are the things that represent numbers? Infinite paths. Regards, WM
From: mueckenh on 16 Nov 2006 14:29 Dik T. Winter schrieb: > In article <1163606442.944751.259900(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > Cantor considered as "Anzahl" which can be determind by "abzaehlen" > > > > simply the ordinal number of a set. > > > > > > But in that quote there is no mention of "Anzahl". There is only mention > > > of "abzählen", and if I understand German well, that means, in general, > > > just like in Dutch, the "process of counting". > > > > That is right. But Cantor has a slightly different understanding. > > That means that he understands under "abzählen" something different? > > > He > > defines: Die kleinste Mächtigkeit, welche überhaupt an unendlichen, > > d. h. aus unendlich vielen Elementen bestehenden Mengen auftreten kann, > > ist die Mächtigkeit der positiven ganzen rationalen Zahlenreihe; ich > > habe die Mannigfaltigkeiten dieser Klasse ins unendliche abzählbare > > Mengen oder kürzer und einfacher abzählbare Mengen genannt; sie sind > > dadurch charakterisiert, daß sie sich (auf viele Weisen) in der Form > > einer einfach unendlichen, gesetzmäßigen Reihe ... darstellen lassen, > > so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe > > steht und auch die Reihe keine anderen Glieder enthält als Elemente > > der Menge. (Collected Works, p. 152) > > Apparently no. What he is describing here is bijections. And what he > is stating is that each countable set can be put in bijection with > another countable set. But that is about cardinals, not about ordinals. Cardinals cannot be determined without ordinals, i.e., without ordered sets. > Sets with the same "Mächtigkeit" can be put in bijection with each other > (this is a generalisation of the above statement). However, the process > of counting ("abzählen") requires an order preserving bijection. Most > ordered sets of cardinality aleph-0 require omega to get such an order > preserving bijection with the ordinals. But there is one exception, > namely the ordered set with ordinality omega. Where do you require omega in the following sets: 2,3,4,..., 1 1,2,3,..., a,b,c,... > > > > There is one set of the > > > first cardinality (N) that can be counted (i.e. the process of counting) > > > without any reference to w. That is potential infinity, and you did > > > agree. And that is still the case in modern set theory. > > > > Cantor would say: "It can be counted into the infinite", later he > > dropped the specification "into the infinite". > > Perhaps. > > > > > This does *not* mean that omega is an element of the set. > > > > > > The quote means that during the process of "abzählen" we need a number > > > of the second class, and so there is an omega-th element (not an > > > element omega). > > > > I cannot agree. > > With what? (Strange enough, in other articles you state that the set N > does contain an un-natural number...) No. I say, and have proved, that IF the set N does actually exist THEN it must contain an infinite number. But this proof shows only that there is no complete set N (because an infinite number cannot be an element of the set N). Cantor could and did not accept that. > > > > > {1,2,3} is the collection of, and a convenient expression to write that > > > > we are talking about, the numbers 1 ,2, and 3. > > > > > > No, the set *containing* the numbers 1, 2 and 3. > > > > The envelope becomes more important than the contents. A characteristic > > of modern times. The publisher becomes more important than the author. > > The director becomes more important than the composer. The trainer > > becomes more important than the football-team. > > Rethoric. Just an observation. > > > > > > Sorry, I do not understand what you write here. The set > > > > > {{1, 2}, {3, 4}} > > > > > has two elements. It's cardinality is 2. And each of the elements > > > > > is a set containing two elements. > > > > > > > > Yes, here you are talking about two unordered pairs of numbers. You use > > > > a convenient way to denote that. Nothing else stands behind the {, } > > > > symbols. In particular N = 1,2,3,... = {1,2,3,...}. That does not > > > > prevent to build a set {{1,2,3,...}, 1} with two elements. > > > > > > And omega = {0, 1, 2, ...} (I use { and } here to denote ordered sets.) > > > > That is not customary. An ordered set is more than a set, because the > > order is added. > > Yes, that is why I state that I use it as such. > > > > So you can not replace omega by 0, 1, 2, ... > > > > Well, that is correct. > > No, you replaced it by 1, 2, 3, ...; but that is also not allowed. I am in doubt. If I say 1,2,3,... then I imply order. Therefore it is even more than to say {1,2,3,...} with the usual meaning of an unordered set. Regards, WM
From: mueckenh on 16 Nov 2006 14:32 David Marcus schrieb: > > > Indeed. If people *object* to an axiom, that is philosophy. > > > > But if people choose a set of axioms, that is what? > > > > > Everyone is welcome to choose their own axioms. > > > > That's mathematics? > > Of course. And if people not decide to use an axiom, that is what? But if people decide not to use an axiom, that is philosophy? > > Would like to do. Please le me know which words are available in your > > universe of discourse. > > I told you several times that the terminology in any modern textbook is > fine. For some reason you do not like this answer. I told you the terminology used in a modern textbook to show that finished infinity is used there. For some reason you do not like to understand it. "Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process (such as N) should not be treated as a finished entity." Regards, WM > What would be something that is "actually infinite"? Read Cantor, he can explain it better than me. > > > e. An "infinite number" is a number other than the natural numbers. > > An "infinite number" would be a number other than a natural number. > Are you agreeing or disagreeing? I am astonished that you cannot understand simplest sentences. You seem to have difficulties with conditional constructs. Should you ever intend to study mathematics be prepared that such constructs will appear quite frequently. > > If an actually infinite set of numbers existed, and if neighbouring > > elements had a fixed distance from each other, then the set must > > contain an infinite number. > Is that a "no" or a "yes"? Read again, simplified: If neighbouring elements have a fixed distance, the answer is yes. If neighbouring elements have not a fixed distance like the rational numbers: the answer is no Regards, WM
From: mueckenh on 16 Nov 2006 14:36 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > 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. > > How do you know this? Do you have any sort of rationale or proof? It > seems such a silly thing to say. Consider: > > Each of the following sequences has a last element: > > 1 > 1 2 > 1 2 3 > 1 2 3 4 > 1 2 3 4 5 > ... > > This sequence does not have a last element: > > 1 2 3 4 5 ... > > This last sequence has three dots on the right. None of the other > sequences do. So, this last sequence is clearly different in some way > from all the other sequences. Yes. We do not know its last element. Perhaps it can change its size. But mathematics does not obey commands. Neither interpreted as a command nor as a magic formula the "..." can create infinity. > "Crank" means someone who makes up pejorative names for people whose > language they do not understand and thinks that definitions which are > explained in many books are "private". So you are a crank? 1) You call others cranks, whose language you don't understand. 2) You do not understand words used by many current text books. BTW: Do you really think it is a proof of your superior intellect if every second word of yours is "sorry don't know"? > How can you do mathematics without axioms? A major purpose of axioms is > to avoid ambiguity. How has it been done over 4000 years? > I can verify I'm using words with the same meanings as other people by > asking them what definitions they are using, then seeing if they are the > same as mine. How could you see that if you don't know what the words in their definitions mean. > We use technical terms to refer to precisely defined mathematical > concepts. Why do you say "we" if you talk about mathematicians? Regards, WM
From: William Hughes on 16 Nov 2006 14:48
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No. The length of the diagonal is the supremum of the lengths > > of the lines (this is easy to show). So if there is no > > longest line the diagonal must be longer than any line. > > The length of the diagonal cannot surpass the length of any line. (This > is easy to show for every matrix.) So, there is a line which has the > same length as the diagonal, i.e., infinite length. > > > > In order to see that, add one element to every column and > > > to every line. Now the order type of any column is omega + 1, the > > > length of the matrix has order type omega + 1, the order type of any > > > line is n+1 < omega, and the width of the matrix has order type omega. > > > The diagonal is a bijection between columns and lines. It des no exist. > > > This shows that the diagonal in the original matrix did not exist > > > either, > > > > No. > > > > Recall, adding one element to every line does not > > change the number of columns. > > Therefore the set of columns is less than the set of lines. There > cannot exist a bijection. The original matrix has no diagonal No. The set of columns is different that the set of lines. However, the number of elements in the set of columns is the same as the number of elements in the set of lines - William Hughes .. |