Cantor Confusion
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From: mueckenh on 9 Nov 2006 13:04 David Marcus schrieb: > > 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? I say: The diagonal cannot be longer than every line. > > 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? The diagonal of actually infinite length omega cannot exist without the existence of a line of actually infinite length omega. The non-existence of the latter implies the non-existence of the former. So we have the result: There is no actual infinity omega > n for n e N. Regards, WM
From: MoeBlee on 9 Nov 2006 13:46 mueckenh(a)rz.fh-augsburg.de wrote: > 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? In a first order theory, such as set theory, a definitional axiom is a sentence in the language of the theory plus one new symbol and such that the sentence satisfies the criteria of eliminability and non-creativity. In many other posts in other threads, I've said what the criteria of eliminability and non-creativity are. And these are discussed in the Suppes book I recommened and in many other basic textbooks in mathematical logic. Even less formally, a definition is a statement that introduces a new symbol (even as a symbol is informally represented by a natural language nickname such as 'is finite' or 'is the union of') but in a way such that any statement that uses the new symbol has an equivalent statement not using the new symbol and such that no statements without the new symbol are theorems with the definition added to the theory unless these statements are theorems even without the definition added to the theory. Even more basically, a mathematical definition MERELY provides a way to ABBREIVATE mathematical expressions, so that a definition does not add to the expressive or deductive power of a theory. And satisfying the criteria of eliminability and non-creativity are sufficient to ensure that definitions are merely abbreviatory in the way I just mentioned. > > > 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. Formal Z set theories are post-Cantor. Formal Z set theories are not answerable to the writings of Cantor. Moreover, the notion of 'limit' as in the limit of a function or 'limit' as in a limit ordinal are rigorously defined in formal set theories. The fact that YOU use these terms in YOUR own way then suggests that we need YOUR definitions of them if you are NOT using them as defined in a formal theory under discussion. > The expressions "number", "limit", and "grow", however, > should be known to the audience of this thread. MoeBlee
From: William Hughes on 9 Nov 2006 13:51 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Every part of the discussion is crystal clear except for WM's > > maunderings which seem more and more to be deliberately obfuscating. > > > > If every line is finite, say, for example, the nth line is of length > > 2*n, and the diagonal is not finite , then the diagonal MUST be longer > > that every line. > > By definition the diagonal of a matrix cannot be longer than every line > of the matrix. No. The length of the diagonal is the supremum of the lengths of the lines. The length of the diagonal will be longer than every line if and only if there is no line with maximum length. As there is no last line, there is no line with maximum length. - William Hughes
From: MoeBlee on 9 Nov 2006 13:51 mueckenh(a)rz.fh-augsburg.de wrote: > 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). Okay, so for you there is no point to draw about your quote other than understanding Cantor's own writings. And, I'll add that in this particular instance, Cantor, as you translated, is not in conflict with the theorem of current set theory that omega is a limit ordinal and the first oridinal that is greater than all natural numbers. MoeBlee
From: Lester Zick on 9 Nov 2006 13:53
On Thu, 9 Nov 2006 00:09:14 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> William Hughes schrieb: > >> > 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. > >The set of natural numbers does not have a maximum element, so what you >just wrote is clearly false. Want to try again? > >> But in order to force you to understand that simple truth, > >Calling something a "simple truth" does not make it more believable, >especially when it is so easy to give an example to show that it is >false. Look who's talking. Is believability your criterion of truth in mathematics? Is "no contrary examples to show that it is false" your criterion of truth in mathematics? I suspect mathematics requires some criterion of truth more substantial than your naive credulity. ~v~~ |