Cantor Confusion
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From: mueckenh on 15 Nov 2006 10:46 Dik T. Winter schrieb: > In article <1163510733.868272.250410(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > 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. > > > > 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? I agree! 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? I agree! > > > 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. > > > > 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. > > > 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." If there are infinite strings elsewhere, then they are in my tree too (in form of paths). If there are no infinite strings, then they are also absent in Cantor's lists. In particular there can be no proof of transcendental strings from those lists. Regards, WM
From: mueckenh on 15 Nov 2006 11:00 Dik T. Winter schrieb: > In article <1163507203.468528.168460(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1163453584.773656.46060(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > > See <J7E0uw.32v(a)cwi.nl>. "... sets of the first cardinality can be > > > counted only through (with the aid of) numbers of the second > > > class..." (Abhandlungen, p 213.) > > > > In German: "während die Mengen erster Mächtigkeit nur durch (mit > > Hilfe von) Zahlen der zweiten Zahlenklasse abgezählt werden können" > > > > > And that was the quote you denied he explicitly stated. He does not > > > state (and I did not write that he did state) that a set of > > > cardinality aleph_0 has an omega-th element, but that can easily be > > > deduced from the above quote. > > > > 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. 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) > 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". > > > p. 213: Durch Umformung einer wohlgeordneten Menge wird, wie ich dies > > in Nr. 5 wegen seiner Wichtigkeit wiederholt hervorgehoben habe, nicht > > ihre Mächtigkeit geändert, wohl aber kann dadurch ihre Anzahl eine > > andere werden. > > Yes, I know that already. I referred to the place where he states that > a set of first cardinality can only be "abgezählt" with numbers of the > second class. And in my opinion "abgezählt" refers to a process, not > to a result (the total number). > > > 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. > > > It cannot be deduced > > that Cantor thought that there was an omegath element. > > I just did. > > > > > A set is its contents - and nothing more! There are no ghosts in > > > > mathematics. > > > > > > Eh? > > > > {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. > > > > 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. > So you can not replace omega by 0, 1, 2, ... Well, that is correct. Regards, WM
From: mueckenh on 15 Nov 2006 11:03 Dik T. Winter schrieb: > In article <1163536221.946682.182290(a)h54g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > > > > 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). > > > > > > This is wrong. There is no ordinal needed to count the elements of the > > > set of all natural numbers. You can count until you weigh an ounce (;-)) > > > but you will never finish. > > > > In this way you can also count the reals without ever finishing. > > Therefore this cannot be the meaning of Cantor's "countable". > > Indeed. But in the case of countable sets you are sure that you reach > each element after a finite number of steps. Not so with uncountable > sets. > > > > Neither the elements you wish to count will > > > be exhausted nor the numbers with which you count. I think this is > > > potential infinity. > > > > Correct. > > > > > On the other hand, when you ask "how many" elements > > > there are in N, you need an infinity (and this is, I think, actual > > > infinity). > > > > Correct. > > > > > But all this hinges quite closely on the semantics of the > > > word "count". If seen as a process, you do not need an infinity; when > > > seen as the result of a process, you do need an infinity. > > > > and, above all, you do not need different infinities. This had been > > known already before Cantor. > > And is still true in current set theory. Set theory does not think too > much about the "potential" infinity, because it is not so relevant for > the theory. On the other hand, other branches of mathematics *do* > consider potential infinities. In analysis you will find nothing but > potential infinities. That is my opinion (and that of Cantor). But I don't believe at many set theorists share it. Most of them do not even know that there could be a difference. Regards, WM
From: Franziska Neugebauer on 15 Nov 2006 11:25 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: >> mueckenh(a)rz.fh-augsburg.de writes: >> > Dik T. Winter schrieb: >> > > mueckenh(a)rz.fh-augsburg.de writes: >> > > > A set is its contents - and nothing more! There are no ghosts >> > > > in mathematics. >> > > >> > > Eh? >> > >> > {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. Gorgeous! F. N. -- xyz
From: William Hughes on 15 Nov 2006 11:51
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1163506214.782141.176790(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > Thank you for your attention. Correction: > > > > > An initial segment of natural numbers includes its cardinal number. > > > > > > > > Further correction. A finite initial segment of natural numbers includes > > > > its cardinal number. > > > > > > Rejected. > > > > Why? Yes, I know why. You think that the complete set of natural numbers > > does contain an un-natural number. > > It must, but cannot. As an un-natural number cannot be a natural > number, this set N cannot exist. > > > I have asked you for a proof based on > > the basic principles. But you simply refuse to do so. You think it is > > the case, so it must be the case. So, please, a proof that N contains > > something that is not a natural number based on the standard axioms of > > set theory. Once you have done so you will have shown an inconsistency > > because it is easy to proof that there is no such number. > > Here is one of several proofs: > > Take the set of natural numbers in form of a list or matrix: > > 1 > 11 > 111 > ... > > This matrix has length omega and width omega. And its diagonal has > length omega. No line has length omega. Therefore the width is larger > than any line. And the diagonal is longer than any line. This is > impossible. 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. > 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. The diagonal of a matrix with the same number of lines and columns exists. If you take a matrix with the same number of lines and columns and add one line and no columns, you end up with something that has more lines than columns. - William Hughes |