Cantor Confusion
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From: Virgil on 15 Nov 2006 16:31 In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > And it does not mean that he is wrong. You cannot even decide this > question without arbitrarily assuming axioms. Your choice is willful. > Therefore your results do not prove anything and are without value. > Cantor created absolute truth, at least where he avoided technical > errors like his diagonal argument. > > > In > > principle no axiom is necessary. But you need a few to have some start > > to work with. > > That's the question. By means of axioms you can produce conditional > truth at most. I am interested in absolute truth. Axioms will not help > us to find it. I don't think we need any axioms. If you have no axioms you cannot produce anything. Nothing begets nothing. Which is what WM has so far begot.
From: Virgil on 15 Nov 2006 16:40 In article <1163603386.223046.315750(a)e3g2000cwe.googlegroups.com>, 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. If we start at 1 and cut the distance to 0 in half successively and endlessly, we never arrive exactly at zero, but we get closer than any positive distance from it. The set of all such distances exists on the real line, and from that set one can form the set of all naturals. > > > 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. This alleged impossibility is an invalid assumption, not true in any number of axiom systems, and without any justification. It is an assumption WM keeps making despite having had it pointed out to him repeatedly as being unjustified and unjustifiable. It is equivalent to assuming that any infinite set is impossible. But WM cannot prove that without assuming it.
From: Franziska Neugebauer on 15 Nov 2006 16:52 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > William Hughes schrieb: >> >> >> Let he original matrix be A. >> >> [...] >> >> > 1 >> >> > 12 >> >> > 123 >> >> > ... >> >> >> >> Something is missing here: >> >> >> >> 1uuu... >> >> 12uu... >> >> 123u... >> >> ... >> > >> > No. >> >> Wrong. You have been discussing matrices. At least William Hughes >> did. Are you both writing at cross purposes? > > I used this suitable word because it allows to speak of lines, columns > and diagonal. _Define_ what you mean! I suspect you are speaking of matrices, lines, columns and the like in default of a reasonable point of view. > If you don't like it, say triangle or structure. I > propose to use "infinite triangle" in order to be clear and to show > that your commentary below fails to show anything. Instead of "square" > we should speak of "equilateral". So we have an Equilateral Infinite > Triangle: EIT. Another misnomer. Cf. my posting in reply to David Marcus. <45550ba7$0$97245$892e7fe2(a)authen.yellow.readfreenews.net> >> > We discuss the question whether an infinite set of naturl numbers >> > requires an infinite number as an element. >> >> There is nothing to debate as this is consistently *not* required >> in current set theory. > > This question is just under discussion. Your sketches are under discussion and your wording. >> > The naural numbers are expressed by the digits of a line. >> >> Perhaps your "expression by the digits of a line" is unapt to >> properly represent the naturals and sets thereof? > > The naturals can be written in unary notation: > >> > >> > 1 >> > 11 >> > 111 >> > ... Set theory is not about your notations. >> If you in advance define what exactly this idea sketch stands for. > > But it is easier to denote indexes like > > 1 > 12 > 123 > ... > > These lines represent the naturals > > 1 > 2 > 3 > ... These sketches are hardly maths. >> > The "u" are not appropriate. >> >> The "u"s are necessary if you are writing about matrices. They >> denote the remaining positions which are not occupied. >> >> >> Usually a matrix m(a,b) is a function of domain A x B (it is >> >> rectangular not a triangle) and some co-domain. Here A equals B >> >> equals omega by definition. A matrix with domain A x B is called >> >> (generalized) square iff A = B. >> >> >> >> "u" stands for undefined (empty), not set. The "triangle" you see >> >> is not the structure of the matrix but its occupancy. >> > >> > We discuss just this triangle as our marix. >> >> Do you want to redefine "matrix"? Twee! > > I used this suitable word because it allows to speak of lines, columns > and diagonal. I want to speak of occupancy. > If you don't like it in this connection, say triangle or > structure. I want to see set theoretical arguments. > I propose to use "infinite triangle" in order to be clear > and to show that your commentary below fails to show anything. Instead > of "square" we should speak of "equilateral". So we have an > Equilateral Infinite Triangle: EIT. Misnomer. Your triangle lacks two vertices. If ever call it "monangle" or just "angle". >> >> Prima facie "Adding x to each column" ("at the end") is not >> >> "possible", since the columns have no end (no last element). >> > >> > If 1,2,3,... has the ordinal number omega and if it is possible to >> > construct 2,3,4,...,1 and if it is meaningful to denote the ordinal >> > number of 2,3,4,...,1 by omega + 1, then your argument fails. >> >> I don't know what this means. But I see that you have not commented >> on the remaining part which nicely explains where you are in error: > > The following matrix is unsuitable to express natural numbers in unary > representation. Untenable assertion. >> > | "Adding x to each row" means simply changing the occupancy of the >> | matrix without changing its domain: >> | >> | 1xuu... >> | 12xu... >> | 123x... >> | ... >> | >> | There is no change in domain "necessary" to "perform" this >> | operation. > > If there is a different effect of adding one element, then the EIT was > not equilateral. Equivocation: "Adding one element" names two different things (changing the occupancy vs. changing the domain). If you chose the matrix-view consequently you would have recognized your error. [...] >> | [...] >> |> No, William, if a matrix has a diagonal crossing every line and >> |> every column, and if one element is added to every line and every >> |> column and to the diagonal itself, then this diagonal again >> |> crosses every line and every column. Briefly: >> | >> | As explained above: "Adding x to each row" is entirely different >> | from "Adding x to each column". > > The different result of increasing the ordinals of columns and lines > is only due to the fact that in the original structure both are > different. As explained: In the matrix-view there is no change of ordinals at all when you "add one to each line". You simply change the occupancy of a sequence member which was not occupied before. Since there is no "last line" you cannot "add one to each column" without extending the matrix structure (domain from omega to omega + 1). > If there were omega columns and omega lines, then increasing by 1 > would yield omega + 1 in both cases. Equivocation. F. N. -- xyz
From: Franziska Neugebauer on 15 Nov 2006 17:12 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> William Hughes wrote: >> > Franziska Neugebauer wrote: >> >> William Hughes wrote: >> [...] >> >> [...] >> >> > The original matrix has the same number of lines and columns, >> >> > however,the lines and columns are different. Doing the same >> >> > thing to the lines and columns does not have the same result. >> >> >> >> You are *not* doing the same thing! >> > >> > Yes and no. In both cases we are "adding a single element", in >> > both cases we are increasing the ordinal represented by a sequence >> > by one. >> >> In the "row case" you increase the finite number of occupied figures >> in each line which does neither change the domain of the sequence nor >> that of the matrix). >> >> In the "column case" you change the matrix structure by extending its >> domain. >> >> > So in one sense we are "doing the same thing". However, >> > adding one element to a sequence without an end means we have >> > to add the element "after" the sequence, while adding one >> > element to a sequence with an end mean we add the element >> > "at" the end. >> >> "Doing the same" applies solely to the wording not to the operations >> acting upon the underlaying mathematical objects. > > If there are infinitely many finite numbers, Does your religious confession require you to eulogize "infinity"? Or do you simply want to render a disclaimer against "infinity"? > then there is a bijection between lines and columns and then line n > has exactly the same properties as the initial segment of the first > column: 1,2,3,...,n <--> 1,2,3,...,n. Whichs properties are you writing about? F. N. -- xyz
From: Dik T. Winter on 15 Nov 2006 19:21
In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > In > > principle no axiom is necessary. But you need a few to have some start > > to work with. > > That's the question. By means of axioms you can produce conditional > truth at most. I am interested in absolute truth. Axioms will not help > us to find it. I don't think we need any axioms. If you want to find absolute truth you should not look at mathematics. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |