Cantor Confusion
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From: mueckenh on 18 Nov 2006 13:23 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > The natural numbers count themselves. Bijection of initial segments of > > > > column and lines > > > > > > > > 1 > > > > 2 > > > > 3 > > > > ... > > > > n <--> 1,2,3,...n > > > > > > > > If there is no infinite number then there are not infinitely many > > > > numbers. > > > > > > This is clearly the point of contention. > > > > > > Consider, N, the set of all natural numbers. > > > By definition N only contains natural numbers. > > > > > > Cases > > > i: There is a largest natural number,n_L, then N={1,2,3,...,n_L} > > > In this case there are n_L natural numbers. > > > > Deleted. > > > > > > ii: There is no largest natural number. We will > > > write this as N={1,2,3,...} (the ... represent only natural > > > numbers). Set N has infinitely many > > > elements. > > > > Please distinguish: > > iia: There is no number counting the elements of N. > > iib: There is a number omega counting the elements of N. > > No case ii is > > there is no largest natural number. > and the set of natural numbers has infinite size. > (i.e.the natural numbers are counted by omega). > > You wish to distinguish > > there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) > > from case iii > > there is no largest natural number. > there are a finite number of natural numbers > > I don't see the distinction (if a set cannot be counted by an > infinite number, what else could it be but finite?) You are so much caught in set theory that you cannot even imagine that there are no infinite numbers, that some sets cannot be counted? >but > let us use the name case iv for > > there is no largest natural number > there in no infinite number which counts N > (i.e. there is no number omega counting the elements of N) > > (it is really not a subcase of case ii) > Your case ii is the only case which considers an infinite set of natural numbers. This infinite set can be potentially infinite are actually infinite. Therefore I chose iia and iib. Edward Nelson, Princeton, says: "There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. We shall not be far wrong if we call these the Platonic (P) and the Aristotelian (A) ways." But if you like to say ii and iv, I will accept it. > > > OK. Knock yourself out. Note however that this case > is not consistent with assuming the axiom of infinity. > The axiom of infinity says that the set N exists. As I remarked already, it does not say that the set has an ordinal number. > > I don't use case iii. > > However, you still have to assume case ii. Starting from the axioms I have to assume case iv. Case ii then may be derived. > > > I use the fact that the diagonal (=bijection, > > d_nn) cannot be longer than every line, because it consists of what you > > have called the line indexes. > > [Actually the lines consist of column indexes, it is the columns > which consist of line indexes. Since both sets consist of > exactly the natural numbers it doesn't really matter] I understood you saying that for 1,2,3,...,n being a line, n is the line index. > > The diagonal contains every column index. No line contains > every column index. Therefore the diagonal is longer > than every line. That assertion is wrong. You cannot show any d_nn the n of which is not in a line. > > Recall, we have assumed case ii, so we > have assumed no largest natural number. > In particular we have no last line. Correct. And we have no complete diagonal. And we have no complete column. > It is trivial to see that if there is no last line then > every line is shorter than the diagonal. > How can you claim that the diagonal cannot be longer > than every line? If you claim that, then you should name an element d_nn for which there is no n in any line. As you cannot show it, the diagonal cannot be longer than every line. So we have two results: 1) The diagonal must be longer than every line. 2) The diagonal cannot be longer than every line. This implies: The diagonal cannot be complete at all. The axiom of infinity is in contradiction with mathematics. > > > For every element of the diagonal we must > > have a line index and, hence, a line. This must hold in the extended > > version too. > > No > > Recall. The extended version involves adding elements (not lines > and columns). Adding one element to every column adds a line, Only if the complete column does exist. But just that is wrong. > adding one element to every line does not add a column. > So if we start with the same number of lines and columns > we do not end with the same number of lines and columns. That should show you that the assertion of an actually infinite set of finite numbers is a self contradiction. > > > This is my point of departure. If you say that the > > diagonal can be longer than every line > > then you say that there are > > more natural numbers (elements of the bijection d_nn) than natural > > numbers (indexes n). > > No. > Yes. d_nn are numbers. > I say that the diagonal consists of the sequence of all natural > numbers. > This means that for any natural number n the diagonal is longer > than {1,2,3,...,n}. Since every line can be written in the > form {1,2,3,...,n} this means that the diagonal is longer than > every line. You say the diagonal has order type omega. No line has order type omega. This prevents a bijection between lines and diagonal. > > > We have an infinite number of lines. We say so. But we forget that this implies to have an infinite line (in case ii). > By case ii there > is no line with infinite index. Thus there is no element > of the diagonal with infinite index. So the set > of elements of the diaongal d_nn is exaclty the > set of natural numbers < omega. > Why does addition of one element yield different results for columns, diagonal and lines? > > The number of lines and number of columns is the same. Why does addition of one element yield different results for columns, diagonal
From: William Hughes on 18 Nov 2006 13:30 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > I claim case iia: There is a potentially infinite sequence N = > > > 1,2,3,..., such that for any n there is n+1 but we cannot recognize or > > > treat all of its elements. In particular we can never complete this > > > set. We can never put it into a list > > > > OK. Knock yourself out. > > I read this several times from you. What does it mean? > > > Note however that this case > > is not consistent with assuming the axiom of infinity. > > The axiom of infinity says that the set N exists. > > But it does not say that it has an ordinal number and a cardinal > number. My case iia is in complete agreement with the axiom of > infinity. > The axiom of infinity says that the set N exists. Your iia says "we cannot recognize or treat all of its elements" The axiom of infinity does not say anything about ordinal or cardinal numbers. However, given that the set N exists and the defnition of ordinal and cardinal numbers, it is easy to see that if N exists it must have both an ordinal and a cardinal number. Ordinal. Every natural number is an ordinal. Every initial sequence of ordinals is an ordinal. (an initial sequence is a set of ordinals, such that if a is in the set every ordinal less than a is in the set). The set of all natural numbers is an initial sequence of ordinals. Therefore the set of natural numbers is an ordinal. Cardinal. Cardinal "numbers" are equivalence classes of sets under the equivalence relation bijection. Since any set has a bijection to itself, every set must be in an equivalence class. So the set of natural numbers has a cardinal number.. - William Hughes
From: David Marcus on 18 Nov 2006 13:31 Lester Zick wrote: > On Thu, 16 Nov 2006 02:28:14 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >Lester Zick wrote: > >> On Sat, 11 Nov 2006 15:53:40 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > The usefulness to other fields is demonstrated via the > > scientific method, not by mathematical proof. > > You mean the empirical method not the scientific method. I think I meant what I said. > So mathematical axioms are to be empirically demonstrated now? I didn't say mathematical axioms are to be empirically demonstrated. I said that other fields demonstrate that mathematics is useful to them using the methods appropriate to those fields. > > Are you saying you don't know what the word > > "proof" means in mathematics? > > I'm saying you can't prove the truth of whatever you say in or about > mathematics. But, do you know what the word "prove" means in Mathematics? It isn't the same as what it means in English. > > A major purpose of axioms is to avoid ambiguity. > > The main purpose of axioms is to provide assumptions of truth without > proof. Why do you think that? > >> Are you going to illustrate the existence of infinites by production > >> of one or more; or are you going to demonstrate the truth of their > >> existence by some alternative means? You posit certain properties and > >> characteristics of things you call "infinites" but don't show they can > >> actually be realized in combination with one another. > > > >Sorry. Don't know what you mean. In particular, I don't know what you > >mean by "illustrate the existence", "demonstrate the truth of their > >existence", "actually be realized". Can you give an example? > > I should give you examples of the examples of infinites I asked you > for? I didn't say that. I said you should give an example to show the meaning of the phrases I quoted. Pick something and "illustrate its existence". > All you do in modern math is prove theorems from assumptions of truth. > Not exactly overtaxing intellectually but there it is. If by "assumptions of truth" you mean axioms, then that is correct. > >Have you read any math books at the junior/senior college level or > >above? > > Apparently more than you. Could be. Which math books have you read? > >> Now I don't say there aren't cranks out there but there is also > >> truth out there and you don't have a clue as to how to get at it. > > > >Why do you think "truth" is relevant to mathematics? > > I gave you the citation. I can't imagine why you think truth isn't > relevant to mathematics. What is the citation? I think I missed it. > >Anyone who learns math is welcome to call themselves a mathematician. > > In other words anyone who learns to agree with you is welcome to call > themselves mathematicians? I didn't say that. > >how it could be that math could be used in "science of all types" if the > >mathematicians are really as you portray. > > Lucky guesses. I never said modern mathematikers, quantum empirics, > and relativists weren't lucky just that they were too lazy or stupid > to figure out the truth of what they were saying. We should all switch from mathematics to playing the lottery. > > It isn't enough to learn the word. First, you have to > > understand the concept. This takes work. > > But you said mathematical definitions are "only abbreviations" and now > alluva sudden you expect people to learn concepts instead? How droll. Sure. There is a difference between a thing and its name. Simply being able to recite a definition doesn't mean you understand it and can use it properly. > >What books on the topic have you read? What courses have you taken? > > Apparently more than you. Could be. Which books have you read and which courses have you taken? > I can't even get you to discuss the truth of what you say. All I hear > about are your assumptions of truth in modern math. I never said that. -- David Marcus
From: mueckenh on 18 Nov 2006 13:34 Franziska Neugebauer schrieb: > >> omega "is" not "the first column". What you may write is, that in > >> your sketch the first column _represents_ omega. > > > > That is a matter of taste. > > You are in error. Set theory under discussion does not deal with > "columns". This column contains all natural numbers in their natural order. It is N, it is omega, written horizontally. > > On the other hand omega represents what before Cantor was commonly > > abbreviated by oo. > > Irrelevant to contemporary set theory. which you, unfortunately, don't know any better than its history. > > > Omega is the first transfinite number. > > Omega is the first transfinite _ordinal_ number. > Please look it up in any modern text book. You will find there that omega is a cardinal number too. > > You need not interpret n as a set (though you can do it). > > In contemporary set theory almost everything is a set. In ZFC everything is a set. Not in every set theory. > > A treatise in which variables ("n") and number symbols ("0", "1", ...) > do not refer to sets is not a treatise _on_ set theory but > a treatise of _application_ of set theory, if ever. Don't mistake set theory with ZF or ZFC. > > It is said to be the number of natural numbers. > > In contemporary set theory it is said that omega is _the_ _set_ of > natural numbers [as Virgil pointed out the _ordered_ set]. The number > (cardinality) of omega is named aleph_0. So it is said that the number > (cardinality) of _the_ _set_ of natural numbers is aleph_0. You are completely in error. The number (Anzahl) of a set is its ordinal number. The cardinal number is something different (Mächtigkeit). But even the cardinal number can be denoted by the least ordinal which can be put in bijection with a set of that number class. > > > If you say that the diagonal can or even must be longer than > > every line, > > I never said that a diagonal (d_ii) i e omega "can" or "must" be longer > than every (finite) line occupancy. What every clear-thinking person > agrees upon is that the cardinality of the sequence (d_ii) i e omega is > greater than the cardinality of (the occupancy of) every single line of > that "list" or "matrix". This is due to the _fact_ that there are only > finitely many occupied memebers in each line-sequence. I agree. But on the other hand, the cardinality of the sequence (d_nn), i. e. omega, cannot be greater than the cardinality of every single line of that "list" or "matrix". This is due to the _fact_ that there are only such d_nn for which an n exists. If omega is a number which can be completed and even surpassed, then there must be at least one line with omega units. > >> possible := exists > >> impossible := does not exist > > > > There is no other interpretation possible, I think. > > Why do you hesitate? It is politeness. I could have said: Everybody whose intelligence is not too far below the average level will understand this interpretation. But I didn't. > > >> > >> Shall I interpret this as "possible := exists, impossible := does not > >> exist"? So why don't you simply use the established terms "exist" and > >> "not exist"? Maliciousness? > > > > Knowledge of literature. Cantor. > > As you have been told quite some time before: Mathematics is no > Zitierwissenschaft (quotation(s)/citation(s) science). Sometimes it is necessary to quote. In particular if you are uninformed but nevertheless refuse to take advice from me. In ZFC the cardinality of a set S is the least ordinal alpha such that there is a bijection from alpha to S. > > > We should pay a bit more respect to the creator of set theory: > > "An mehreren Stellen meiner Arbeit werden Sie die Ansicht > > ausgesprochen finden, daß dies unmögliche, d. h. in sich > > widersprechende Gedankendinge sind, ..." > > I don't discuss Cantor's remarks. > Then, please stop to complain when you do not understand my expressions. I do discuss Cantor's remarks. > > You claim that the diagonal of a matrix can be longer than > > every line. > > This is your wording not mine. It is not your wording, because you like to veil your inconsistencies, but it is your opinion. > > > As the diagonal is defined to consist of the ends of terms > > of lines, this claim is easy to conradict. > > "ends of terms of lines" is your wording not mine. It is not your wording, because you prefer to veil your inconsistencies, but it is your opinion. > > > In order to defend your claim that there are infinitely many finite > > numbers, > > In the lack of an effective offense I don't have to defend but to inform > you of facts. Why then do you complain about the wording "the diagonal of a matrix can be longer than every line"? > > > you could simply say: > > > > "There are more natural numbers d_nn than natural numbers n." > > Your wording not mine. Can you understand mathematical symbols? Can you understand d_nn <--> n? > > > This sentence is as true as the sentence: The diagonal (d_nn) of a > > matrix can be longer than every line n. But it shows that there is no > > point in further arguing on a logical basis with sound arguments. > > Have _you_ ever been arguing on a logical basis with sound arguments? Yes, but that is very different from what you erroneously consider to be logical and sound. Regards, WM
From: stephen on 18 Nov 2006 13:31
David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Lester Zick wrote: >> On Thu, 16 Nov 2006 02:02:49 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >Lester Zick wrote: >> >> "Provability" of what pray tell? If you're not concerned with proving >> >> the truth of what you say in mathematics exactly when are you not >> >> discussing philosophy every time you say anything in mathematics? >> > >> >Do you really not know the mathematical meaning of the word "prove"? If >> >so, I (and others) could try to explain it to you. But, if you are just >> >being argumentative, we won't bother. >> >> What is it you think you're proving? > Does that mean you don't know the mathematical meaning of the word > "prove"? It isn't the same as the English meaning. >> >Please give a specific example of something that you think is absurd or >> >a contradiction. I don't know what you mean by "containment of sets and >> >subsets". >> >> Well as I recollect Stephen seems to think infinite sets are proper >> subsets of themselves. > Are you sure that is what Stephen thinks? I see Lester has resorted to lying. This is all part of his standard pattern. He really is pathetic. It is amusing to see how increasingly pathetic he becomes. Stephen |