Cantor Confusion
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From: Virgil on 9 Nov 2006 00:12 In article <1162986270.218701.275100(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > I answered that already. If you define 'the width', then it turns out > > to be equal to omega, which is just what the length of the diagonal is. > > The diagonal is assumed to exist such that each of its digits exists, > actually. This is established by the mapping on a column. But it cannot > be established by the mapping on any line. Why should the diagonal have to be mapped onto one line when it is not constructed from one line? WM seem to have all sorts of peculiar requirements for things that need not be required for anything except his own squirrelly views of things. > > I answered that already. And the cardinality of the diagonal is not > > assumed, but is proven, to be omega. > > It is *assumed* by stating the axiom of infinity. Without this > assumption the length was not omega. The axiom of infinity has not been shown to cause any problems within ZF or NBG, and WM has not produced any /system/ in which it does not hold.
From: Virgil on 9 Nov 2006 00:15 In article <1162988209.235780.201370(a)m7g2000cwm.googlegroups.com>, "William Hughes" <wpihughes(a)hotmail.com> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > > If he considers omega to be not the maximum but only the supremum of > > > > the set of lines, then we agree that actual infinty does not exist. > > > > > > No. > > > > > > You seem to be saying that: > > > > > > if the set of lines does not contain a line of maximum length > > > then the set of lines does not exist. > > > > > > However, my claim is that: > > > > > > the set of lines does not contain a line of maximum > > > length > > > and the set of lines exists. > > > > That means: All its elements exist? > > > > > > It is not true that a set cannot exist unless it has a maximum element. > > > > It is true, at least if all of its elements do exist. But in order to > > force you to understand that simple truth, consider the bijection > > between lines and columns. All lines do exist, all columns do not > > exist. > > > > It is necessary to distinguish between the lengths of the > lines and columns and the indexes of the lines and columns. > > Let the set of line indexes be LI. > Then LI is just |N. For every natural number n we have a line > n. There is no last line so there is no line infinity. > > Let the set of column indexes be CI. > Then CI is just |N. For every natural number > n there is a column n. > > Let the set of column lengths be CL > Then CL has only one element, aleph_0. > The supremum of CL (also the maximum) is > aleph_0. > > Let the set of line lengths be LL. > Then LL is |N. LL does not have > a maximum. However it does have > a supremum, aleph_0. > > The bijection is between LI and CI, not between > LL and CL. > > - William Huhges Quite clear and to the point. It sinks WM's silly arguments totally, though WM will no doubt remain willfully blind to his own errors.
From: David Marcus on 9 Nov 2006 00:13 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > 1 > > > > > 11 > > > > > 111 > > > > > ... > > > > > > > > > > The length of each column is omega. > > > > > > > > I assume that we are using ZFC as our logical system. > > > > > > > > Assuming by the "length of each column" that you mean the number of 1's > > > > in each column, then the number of 1's in each column has cardinality > > > > aleph_0. So, what you wrote is essentially correct. > > > > > > > > > The length of each line is less than omega. > > > > > > > > Each line has a finite number of 1's. Finite cardinality is less than > > > > aleph_0. So, what you wrote is essentially correct. > > > > > > > > > The length o the diagonal is less than omega. > > > > > > > > The number of 1's in the diagonal is aleph_0. So, what you wrote is > > > > false. In fact, I have no clue what you could possibly be thinking that > > > > would lead you to make such an obviously incorrect statement. The length > > > > of the diagonal is clearly the same as the length of the first column, > > > > and you just wrote above that the length of each column is omega. > > > > > > The length of the diagonal is clearly not more than the length of any > > > line. > > > > Do you disagree that the length of the diagonal is the same as the > > length of the first column? > > No it is the same. I distinguish beween lines and columns. OK. The length of the diagonal is the same as the length of the first column. > > As for your statement that "the length of the diagonal is clearly not > > more than the length of any line", the length of the first line is 1, > > and this is less than the length of the diagonal, so your statement as > > written is false. > > Maybe that my use of "any" was wrong. Please replace it by "every": the > length of the diagonal is clearly not more than the length of every > line. So, you are saying that there is some line whose length is greater than or equal to the length of the diagonal. Is that correct? Since the length of the diagonal equals the length of the first column, you are also saying that there is some line whose length is greater than or equal to the length of the first column. Is that correct? -- David Marcus
From: mueckenh on 9 Nov 2006 05:11 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > > Here is my translation: It is allowed to understand the new number > > > > omega as limit to which the (natural) numbers n grow, if by that we > > > > understand nothing else than: omega shall be the first whole number > > > > which follows upon all numbers n, i.e., which is to be called larger > > > > than each of the numbers n. > > > > > > That's not a definition. It is just a remark. > > > > It is a definition. You should not try to judge about topics which you > > must reject because you do not understand them. Please learn: A > > definition is an explanation in other words. Nothing more can be done. > > No, YOU are IGNORANT of the subject of mathematical defintions. A > mathematical definition is NOT just any explaination in words. What is it then, according to your understanding? > > > Would you understand: "The new number omega is the limit to which the > > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > > because you don't know the meanings of "number", "limit", and "grow"? > > No, it's just that you haven't given MATHEMATICAL definitions of them. I quoted Cantor. You cannot expect that I give a mathematical definition of a notion which, in my eyes and considered objectively, is nonsense. The expressions "number", "limit", and "grow", however, should be known to the audience of this thread. Regards, WM
From: mueckenh on 9 Nov 2006 05:13
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > > > > Here is my translation: It is allowed to understand the new number > > > > omega as limit to which the (natural) numbers n grow, if by that we > > > > understand nothing else than: omega shall be the first whole number > > > > which follows upon all numbers n, i.e., which is to be called larger > > > > than each of the numbers n. > > > > > > Okay. I don't see any problem with that. Would you please refresh the > > > context by saying what point it is that you draw from that quote? > > > > Cantor's first book (general paper): Grundlagen einer allgemeinen > > Mannigfaltigkeitslehre (Leipzig 1883). Foundations of a general set > > theory. > > § 11, showing how one is lead to the *definition* of the new > > numbers... (Es soll nun gezeigt werden, wie man zu den Definitionen der > > neuen Zahlen geführt wird und auf welche Weise sich die natürlichen > > Abschnitte in der absolut-unendlichen realen ganzen Zahlenfolge, welche > > ich Zahlenklassen nenne, ergeben.) > > I asked you what point you draw from the quote that you translated. Es soll nun gezeigt werden, wie man zu den Definitionen der neuen Zahlen geführt wird It is now to be shown how one is lead to the definition of the new numbers. And this definition, given by Cantor and translated by myself is given above. There is no point to draw but only to understand Cantor's *definition* (or not). Regards, WM |