Cantor Confusion
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From: Dik T. Winter on 10 Nov 2006 22:05 In article <1163180340.117151.181950(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > There is no difference between enumerating all the rational numbers in > > > the well-known scheme, starting at the corner of an infinite square > > > matrix > > > > There is. You can assign numbers to edges that terminate at nodes. And > > *that* numbering is different. > > I can assign numbers to edges that start at nodes. I can also assign > numbers to nodes. Yes. In the final tree what edge starts at 1/2? As I stated already some time ago, there are two possible trees to build. One of the trees has edges that terminate at a node, the other tree has edges that start at a node and do not terminate. When completing, neither tree has 1/5 as a node. You should really consult Conway how that is resolved. > > It numbers only those edges that are finitely > > far away from the root. But that way you do not number all edges in the > > infinite tree, because that contains edges that are *not* finitely far > > away from the root. > > That is an interesting remark. The paths are isomorphic to the binary > representations of real numbers. So you claim that there are positions > in the binary representation of the real numbers that are infinitely > far from the decimal point. Wrong, again. You are still obsessed with the property that an infinite set of numbers must contain an infinite number. Pray, give a mathematical proof of that property. > > There is a difference, see above. You can only number edges that are > > finitely far away from the root, but in that way you will only number > > edges that terminate at nodes finitely far away from the root. As in > > the edges there are nodes infinitely far away from the root you will > > never number the edges that terminate there. > > Let us enumerate the nodes. Yes, go ahead. > > The situation with the rationals is quite different, because in the > > matrix *each* rational is finitely far away from the root. > > Each node is finitely far from the root. (Does Conway really tell what > you reproduce here?) Well, my only advise is, read it. > I have experienced worse opinions. But let us finish with the polemics > (if possible), because we are at the most important point. Please think > over your argument: > 1) Do you say that the nodes cannot be enumerated? Depends on how the tree is built. If it is built with terminating edges the answer is no, if it is built with non-terminating edges the answer is yes. > 2) Do you agree that this implies: There are bit positions infinitely > far from the decimal point (or how this point may be called for binary > numbers). No. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 10 Nov 2006 22:15 In article <1163182382.226922.297900(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > They are. But in mathematics comparing numbers comes in quite late in > > the process of definition. > > In mathematics counting and comparing numbers comes before most other > things. Not because I think so, but because mathematics developed this > way. In that case you are talking about a different kind of mathematics than what I am talking about. > > Where are they? That there are "limit ordinals" does not mean there are > > "limits" in set theory. > > Why are they called so? Ask the people who coined those names. > > Since Cantor quite a bit has changed. Limits are *not* part of set > > theory, they belong to topology and other things build on set theory. > > Experience has shown that practically all notions used in contemporary > mathematics can be defined, and their mathematical properties derived, > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > include limits? Darn. Do read. It does include limits, but not in the branch of set theory. In that branch the term has not been defined, and so is meaningless. If you want to use such a term in set theory, you have to define it in terms of set theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 11 Nov 2006 03:43 Dik T. Winter schrieb: > Indeed. Wolfgang Mueckenheim appears to think that Cantor's writings are > still all true in modern mathematics. I think that much of his set theory is wrong but more of it is true than of modern set theory. > The first falseness is his assumption that every set can be well-ordered. This assumption is as wrong as the official assumption made by Zermelo. > We all know now that that can not be proven from first principles. According to modern mathematics nothing can be proven from first principles. Cantor considered well-ordering as a first principle, Zermelo introduced it at a first principle = axiom. Cantor was wrong, Zermelo was right? At the most in the ridiculous axiom faith of "modern mathematics". > His second falsehood is when he states that a set of first cardinality > (meaning sets of cardinality aleph-0) can only be counted with use of > numbers of the second class (meaning omega and larger). And I think that > especially this quote has lead Wolfgang Mueckenheim astray. Also see > my discussion about this quote with Dave Seaman. The falsity is apparent > if you realise that quote means that every set with cardinality aleph-0 > has an omega-th element. No explicitly stated but implied. Cantor explained it (as I posted recently and repeat it here): Zum Beispiel betrachten wir die Menge aller endlichen positiven ganzen Zahlen, so ist sie in der natürlichen Folge 1, 2, 3, ..., nu... eine wohlgeordnete Menge und hat in dieser Ordnung gedacht die Anzahl: omega. Schreibt man sie aber in der Ordnung (n + 1), (n + 2), ... (n + nu), ....1, 2, 3, ...., n, so hat sie nun die Anzahl omega + n. In der Ordnung 2, 4, 6, ..., 2nu, ..., 1, 3, 5, ..., (2nu + 1), ... hat dieselbe Menge (nu) die Anzahl 2omega u.s.w. u.s.w. Jede Menge von der Mächtigkeit erster Classe ist abzählbar durch Zahlen der zweiten Zahlenklasse (II); und zwar läßt sich jede Menge von der Mächtigkeit erster Classe in solche Succession (als wohlgeordnete Menge) bringen, dass ihre Anzahl mit Bezug auf diese Succession gleich wird einer beliebig vorgeschriebenen Zahl alpha der zweiten Zahlenclasse. (Georg Cantor in a letter to Mittag-Leffler, Dec. 17,1882) There is no element omega. But, of course, if the set is to be counted, then there must be a number omega following after all natural numbers. This is the error. Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen nu zustreben, wenn darunter nichts anderes verstanden wird, als daß omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt. In the examples above, we have no omega. Introducing it in fact as number which follows on all (even) natural numbers, we get 2, 4, 6, ..., 2nu, ..., omega, 1, 3, 5, ..., (2nu + 1), ..., 2omega That is false. I think this has to do with the problems of the conceptions of potential infinity vs. actual infinity that played an important role in mathematics before the early 1900's. Moreso because there were also religious issues. So Cantor crossed the border (after consultation with his religious leader and in strong opposition from him). The actual infinite was impossible (and unreligious) before that time, but he introduced actual infinity. No. There are hints on the actual infinite in the holy bible and by Saint Augustin. Cantor corresponded with Cardinal Franzelin about that. The Cardinal did not oppose to the idea. He agreed that God will know Cantor's numbers "if they are not contradictory". The Cardinal only disagreed with Cantor's "proof" of the actual infinite. Because Cantor assumed that God had been forced to create it. > And > so, apparently, he thought that if a set was actually infinite, it should > have an actually infinite element, or somesuch. It should have an integer number counting its elements. Of course this number had to appear in the sequence of numbers if it was a number. But where? Obviously after all natural numbers. But then, what is the number of 1,2,3,...,omega ? It cannot be omega + 1, because, according to his equation above, 2,3,4, ....,1 has already the number omega + 1. Therefore omega cannot appear in the sequence. So we have the irrevocable dilemma: 1) omega is the number of countable many numbers like 1,2,3,... or 7,8,9,... and as such a single term in the sequence following the counted terms like 1,2,3,...,omega or 7,8,9,...,omega 2) omega + 1 = 2,3,4,...,1 and therefore omega cannot not be a term in the sequence but omega is the first part of it, omega = 2,3,4,... This dilemma is what I have been trying to explain for years now. Regards WM
From: imaginatorium on 11 Nov 2006 04:09 David Marcus wrote: > Franziska Neugebauer wrote: > > David Marcus wrote: > > > > > [...] you are working with an infinite triangle [...] > > > > ,----[ http://en.wikipedia.org/wiki/Triangle ] > > | A triangle is one of the basic shapes of geometry: a polygon with > > | three vertices [...] > > `---- > > > > Are there really three vertices in WM's "triangle"? > > No. But, I don't think an "infinite triangle" needs to be a triangle. > However, I'm open to suggestions for what to call it. You could call it a "two-sided triangle". This might turn out to be useful in a quiz some day. Brian Chandler http://imaginatorium.org
From: Franziska Neugebauer on 11 Nov 2006 07:14
David Marcus wrote: > Franziska Neugebauer wrote: [...] >> I would prefer discussions either with precisely defined (formalized) >> "infinite triangles" or better without all these words borrowed from >> Cantor/geometry/physics. > > In that case, if you discuss anything with WM, you will be > disappointed. Depends on your expectations. You will find a plethora of absurd constraints for thinking. > He has his own defintion for the word "definition". He does not at all enact the notion of definition. > By "infinite triangle", I meant a function with domain {(n,m)| n,m in > N and m <= n} and range N. OK. So you are writing about abstract entities like functions and domains whereas WM does not. In WM's view a /notation/ like 1 1 2 1 2 3 ... *is* the object under consideration whereas for you it is merely an illustration or reference the abstract entity. What WM calls "diagonal" means the geometrical object contained in that notation either on screen or written on paper. Mathematically "diagonal" simply means the sequence of f(i,i) i e N which by definition has cardinality |N|. WM does not belong to the "factinista". He is ignorant about the fact that nowadays mathematics is a formal science in the first place. F. N. -- xyz |