Cantor Confusion
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From: Lester Zick on 9 Nov 2006 17:03 On 9 Nov 2006 10:46:14 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >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. So, Moe(x), are we to understand that mathematical definitions are merely abbreviations? Personally I find it remarkably difficult to see what the domain of discourse amounts to if definitions are just abbreviations. Yet it's still a little perplexing what the domain of discourse might be. You seem to be able to define various properties as "def(x)=" and so forth but you don't seem quite so able to define the x. Perhaps there is some poetic interpolation involved? ~v~~
From: David Marcus on 9 Nov 2006 18:36 Virgil wrote: > In article <1163067640.060706.37510(a)k70g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > The length of 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. > > > > > > If every line is finite but at least of length equal to is line number, > > > then even more clearly, the diagonal is longer that every line. > > > > That is outright nonsense. > > How is it nonsense when that diagonal contains a position for every n to > say that it is longer that any line of length less than n? > > It is outrageous nonsense to deny it. > > > Its acceptance by set theorists makes set > > theory suspicious as a theory without any value. > > Its denial by EB makes EB a laughing stock. I think you mean "WM", not "EB". > > > So that WM could hardly be more wrong if he tried. > > > > >From the side of a set theorist I consider that as a compliment. > > Then EB will remain in ignorance by his own choice. -- David Marcus
From: David Marcus on 9 Nov 2006 18:50 mueckenh(a)rz.fh-augsburg.de wrote: > 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. I seriously doubt that Cantor considered it to be a definition. If he did, he wouldn't have said, "It is allowed to understand". Regardless, it is not a definition by modern standards. Nor, is it the defintion of omega in modern mathematics. And, since the set of mathematicians who know the mathematical meaning of the words "limit", "number", "grow" is probably empty, your final sentence seems vacuous. -- David Marcus
From: David Marcus on 9 Nov 2006 19:03 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: > > > > > 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. > > > > How would you know, since you never explain anything you say? > > I do explain. But a certain basic vocabulary is necessary to > understand. > > > > > 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. > > > > So, "finished infinity" isn't even in the book you reference. Kind of > > silly for you to refer us to a book to look up what a phrase means when > > the book doesn't even use it. > > Do you always repeat only full sentences you read in a book? > Did you never try to learn substituting words in sentences? > > If 2 + 3 = x you can be sure that x + 5 = 10, even if that is not > written down in a book. Now apply this to "finished entity" and "a > collection of objects produced by an infinite process" Your changing "finished entity" to "finished infinity" is a bit of a stretch. Regardless, Hrbacek and Jech are clearly making a philosophical comment, not discussing mathematics. So, it remains true that "finished infinity" does not have a mathematical meaning. If you wish to use the term in a mathematical discussion, it is incumbent on you to define it. -- David Marcus
From: David Marcus on 9 Nov 2006 19:06
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus wrote: > > > > > That's a wrong definition of the word "definition", at least in > > > > Mathematics. > > > > > > As you believe? > > > > As mathematicians believe. A definition, at least in the mathematical > > sense, is merely a form of abbreviation. It allows one to say more > > briefly what could be said without it but at greater length. > > Correct. An expression, a long one or a short one, is explained in > other words, more briefly or more detailled, respectively. No, that is still wrong. That's not what the word "definition" means in Mathematics. -- David Marcus |