Cantor Confusion
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From: mueckenh on 15 Nov 2006 15:39 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. 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. > > > 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. > > > 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 > > ... > > 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 .... > > > 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. If you don't like it in this connection, 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. > > >> 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. > > | "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. Then there was not a bijection between lines and columns possible, or also: between lines and initial segments of the first column: 1 2 3 .... n and 1,2,3,...,n > | [...] > |> 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. If there were omega columns and omega lines, then increasing by 1 would yield omega + 1 in both cases. As this is not the case, we see (should be able to see) that there were less than omega columns in the EIT. > You are comparing apples and oranges. I proved that they are not of same taste and colour. Yes, that is my purpose. Regards, WM
From: mueckenh on 15 Nov 2006 15:41 William Hughes schrieb: > Franziska Neugebauer wrote: > > William Hughes wrote: > > > > >> You want to say that A has a diagonal, but addition of one element to > > >> each line, each column and the diagonal eliminates this property? > > > > > > Yes, because adding one element to each column changes the number > > > of lines, but adding one element to each line does not > > > change the number of columns. > > > > What did I tell you!? > > Yes, but the discussion has reached the point where > the distinction between adding an element to an infinite > sequence and adding an element to a finite sequence > can be made clear. If there is a bijection between lines and columns, then line n has exactly the same properties as the initial segment of the first column: 1,2,3,...,n <--> 1,2,3,...,n. Then and only then increasing the ordinal by one (adding one element to every line and to every initial segment of the first column) has the same effect in both cases. As it does not have the same effect. This shows that there are not infinitely many finite numbers. > My contention is that an earlier > attempt to clarify this would not have been productive. > (not to say that this attempt will be productive, just > that it is my opinion that it has a better chance of > being productive). > > You may disagree, but it's my time that I'm wasting. The time you have been wasting to learn set theory? Regards, WM
From: mueckenh on 15 Nov 2006 15:42 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, 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. Then increasing the ordinal by one (adding one element to every line and to every initial segment of the first column) has the same effect in both cases. As it does not have the same effect. This shows that there are not infinitely many finite numbers. Regards, WM
From: Virgil on 15 Nov 2006 15:57 In article <1163583873.277287.152670(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > There is no bijection between lines and columns. > > Regards, WM There is outside of WM's delusional world.
From: Virgil on 15 Nov 2006 16:03
In article <1163584329.163361.245230(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Then let's construct some Cantor list, take any line, for instance the > first, and add it to the end of the list as a last line. > > a_21,a_22,a_23,... > a_31,a_32,a_33,... > a_41,a_42,a_43,... > ... > ... > ... > a_11,a_12,a_13,... > > And abracadabra, all uncountability proofs vanish. Not for those who understand what is going on. The diagonal construction procedure is a function defined on the set of all lists. Every time you change the list, the function automatically changes the diagonal value to match it. |