Cantor Confusion
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From: Ross A. Finlayson on 9 Nov 2006 06:09 Virgil wrote. Virgil: you're a known liar. Besides that: you're uneducative, ineducative. Basically, that makes you a detractor, Virgil. So, how's your challenge? There's only so far you can change. Ross
From: mueckenh on 9 Nov 2006 06:11 William Hughes schrieb: > mueck...(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > Oh no? The lengths of the columns of the list > > > > > > > > 0.1 > > > > 0.11 > > > > 0.111 > There are aleph_0 lines and aleph_0 columns. Both infinities > are exactly the same. Why then is there a column with aleph_0 terms but no line with aleph_0 terms? Is that really *exactly* the same? Well, probably in set theory where one does not look too sharp. > > The fact that no line contains all the columns does not change > the number of columns. > > The fact that no line contains all the columns and the first column > contains all the lines does not change either > the number of columns or the number of lines. That is an assertion. But it is wrong. > > > > But why is it not different from the columns? > > It is different from the set of columns in any line. Of course it is different from the set of columns in any line. Because this set is the set of terms of a line. > > It is not different from the set of columns. That is in question. For this purpose we should consider the terms in a column and the tems in a line. There is always a difference. But according to the symmetry of a square matrix, there should be no difference. > > This supremum is a thing which is not realized by the number of columns > > but which is realized by the numer of lines (if the notion of finished > > infinity is correct). What may be the reason for this difference? > > > > Construction. So what? The supremum of a set A may or may > not be a member of A. If it is not a member, then A' has one member less than the otherwise same set A where it is a member. Therefore A and A' are different, aren't they? > In the case of the set of column lengths, C, > the supremum is a member of C. In the case of the set of line lengths, > L, the supremum is not a member of L. However, the supremum of > L is equal to the supremum of C (two different sets can have the > same supremum). Two sets which are distinct by one element, are two different sets. > > > > Because, by construction, every column is infinite and no line is. > > > Why do you insist this simple fact is a contradiction? > > > > Because in natural numbers we have n = n, e.i., the n-th number has > > size n. If you transpose the matrix, then nothing can change. But in > > our matrix, we have a change. This is not possible for natural numbers. > > > > Yes we have a change. The sets C and L change. However, since > the supremums of C and L are identical, the supremums do not change. But after the change the lines have infinitely many indexes while the columns have only finitely many. What about the iterpretation as finitely many infinite numbers? > > Maybe. But a diagonal cannot exist in a domain where no lines reach. > > Correct, however there is no such domain. To say that the > diagonal has aleph_0 1's does not mean that the diagonal has > an index aleph_0. It does not. The diagonal only has finite indexes. But it has infinitey many indixes, while no line has infinitely may indexes. So there is a lack of second indexes n in d_nn. > Given any finite index there is a line that reaches that index. Of course. Could it be that everything stating the existence of infinitely many lines is nonsense, considering the fact that infinitely many terms do not exist in any line? Regards, WM
From: mueckenh on 9 Nov 2006 06:29 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > The set of lengths of columns, C, consists of the single element > > > omega. > > > > This is a maximum taken by every length of a column. > > > > > The set of widths of lines, W, is the set of all natural > > > numbers. > > > W has more elements than C, but every element of W is > > > smaller than every element of C. > > > > Correct. > > > > > > The supremum of W is omega. The width of the > > > matrix is the supremum of the set of widths of the lines, i.e. > > > omega. Thus the width of the matrix is equal to the height of the > > > matrix. > > > > But this supremum is not taken. > > > > The diagonal connects width and length by a bijection. The element d_nn > > of the diagonal maps the n-th line on the first n elements of the first > > column. As long as n is a natural number, this is no problem. Only for > > aleph_0 the diagonal has to map a not existing maximum on an existing > > maximum. This is a hard task. > > No. You are confusing the length of the columns with the set of line > indexes contained in each column. These are not the same. Explain how in a square matrix (squarity proven by the existence of a diagonal) the columns can be longer than the lines for *every* line and every column. > > The set of line indexes contained in the first column is |N. > The length of the first column is the size of |N, aleph_0. > > The set of column indexes is |N. > > The bijection defined by the diagonal connects the line indexes > with the column indexes. Finding a bijection between |N and |N > is not hard. Of course not, because you can do that for finite indexes only. But finding that there is a column with aleph_0 indexes while there is no ine with aleph_0 indexes shows that the idea of the existence of aleph_0 is false. > > In other words, the fact that there are aleph_0 lines does not mean > that there is a line with index aleph_0. Such a line would have > to be the last line, however, there is no last line. If there were aleph_0 lines with aleph_0 a (non-natural) number, then we would need a last line. Therefore "the fact" that there are aleph_0 lines is not a fact. It is simply the result of quantifier exchange to say E a A n : n < a instead of A n E a : n < a. Regards, WM
From: mueckenh on 9 Nov 2006 06:34 Randy Poe 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. > > > > > > That's a wrong definition of the word "definition", at least in > > > Mathematics. > > > > As you believe? > > It's not a matter of belief. A definition DEFINES. It states the > meaning and that has to be done in other words, if the meaning of the original wording is unclear. Exactly this is done by Cantor's definition given above: "It is allowed to understand the new number omega as limit to which the (natural) numbers n grow". This is a definition, at least for those mathematicians who know what "limit, number, grow" means. Regards, WM
From: mueckenh on 9 Nov 2006 06:46
William Hughes schrieb: > 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. There are two sets the bijection of which has not yet been considered but sould be: A = The set of lengths of initial segments of the (first) column. B = The set of lines. Here we can set up a bijection, expressing the lines by the natural indexes of the elements: 1 1 2 1,2 3 1,2,3 ... ... n 1,2,3,...,n ... ... omega 1,2,3,... which shows that omega or aleph_0 does not exist. Regards, WM |