Cantor Confusion
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From: William Hughes on 8 Nov 2006 05:41 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. 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. 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. - William Hughes
From: mueckenh on 8 Nov 2006 05:52 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? > > 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"? > > Sure, I understand it as a remark. But, it still isn't a definition or > theorem unless the other words are defined. The problem is that words cannot be defined unless other words are known. As it seems you don't know words like "number", "limit", and "grow". Therefore it is impossible to define something so that you can understand it. Please note: In mathematics definitions are not necessary at all. All expressions can be given using some primitives. (Therefore your belief expressed above is wrong.) But without those primitive words no discurse is possible. Regards, WM
From: Randy Poe on 8 Nov 2006 06:01 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 of a term. A rambling paragraph which fails to say "the meaning of X is Y" is not a definition. - Randy
From: mueckenh on 8 Nov 2006 06:01 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > Modern mathematics does not use the term "finished infinity". If you > > > wish to use it, please define it. > > > > Modern mathematics assumes that there are sets which are larger than > > infinite sets. That is the same as to say after one finished infinity > > we consider the next infinity. > > I asked you for a definition of "finished infinity". Is what you wrote > supposed to be the definition? Normally, a definition should start out > something like "Finished infinity is ... ". It is tedious to explain things to you. Please consult a book. Hint: "Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process (such as N) should not be treated as a finished entity." (Karel Hrbacek and Thomas Jech: "Introduction to set theory" Marcel Dekker Inc., New York, 1984, 2nd edition.) Now substitute "entity" by the most significant property of the set N. Regards, WM
From: mueckenh on 8 Nov 2006 06:11
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. Regads, WM |