Cantor Confusion
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From: David Marcus on 18 Nov 2006 15:36 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > Take the set of natural numbers in form of a list or matrix: > > > > > > 1 > > > 11 > > > 111 > > > ... > > > > > > This matrix has length omega and width omega. And its diagonal has > > > length omega. No line has length omega. Therefore the width is larger > > > than any line. And the diagonal is longer than any line. This is > > > impossible. > > > > No, that is very possible. > > If you assert that there is no line longer than the diagonal, you have > good reasons, which can be proved. > > If you assert that the diagonal can be longer than any line, then you > have no reasons, because the diagonal consists of line elements and > cannot be where no line is. So your second assertion is outside of > logic and outside of any mathematics. Therefore I am not willing to > discuss this topic further. Does that mean you aren't going to post any more? > The correct result: There must be at least one line which is exactly as > long as the diagonal. There must be an infinite natural number. That is > impossible. Therefore, there is no actually infinite number of natural > numbers, -- David Marcus
From: Franziska Neugebauer on 18 Nov 2006 15:38 mueckenh(a)rz.fh-augsburg.de wrote: > 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. Set theory still does not deal with "columns". You talk about "columns". >> > 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. Still irrelevant. >> > 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. Anyway, omega is not a /natural/ number. >> > 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. I don't want to debate about the axioms of ZFC being sets. The point is that you want to talk about "columns" which neither ZFC nor any other contemporary set theory is about. >> 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. A theory of _sets_ is not a theory of _columns_. >> > 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. "Anzahl" is your (or a historical person's) wording not mine. I can't spot any error in my wording. > The cardinal number is something different (M?chtigkeit). What do you want to posit? > 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. Can you rephrase that? >> > 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". Could you rephrase (formalize) what you mean by "cannot be greater"? > 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, Define "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. Do you understand your interpretation? >> >> 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. I don't need any advice from you. > In ZFC the cardinality of a set S is the least ordinal alpha such that > there is a bijection from alpha to S. What is the cardinality of 0. {} 1. { n | n < k } k e omega? 2. omega? 3. omega u { omega }? >> > We should pay a bit more respect to the creator of set theory: >> > "An mehreren Stellen meiner Arbeit werden Sie die Ansicht >> > ausgesprochen finden, da? dies unm?gliche, 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. Then please do not claim that ZFC is "contradictory". The "C" in ZFC is not for "Cantor". >> > 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. My opinion is A (n e omega & |{0, 1, 2, ..., n}| < |omega|) >> > 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. My opinion is |(d_nn) n e omega| = |omega| >> > 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"? Since even after replacing "can" by "exists" _your_ sentence does not make sense to me. There is no "can" in set theory. >> > 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? Writing down a bunch of symbols does not mean that you create
From: David Marcus on 18 Nov 2006 15:41 stephen(a)nomail.com wrote: > 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: > >> >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. And, such a silly lie. What could he hope to gain? -- David Marcus
From: Franziska Neugebauer on 18 Nov 2006 15:43 David Marcus wrote: > Franziska Neugebauer wrote: [...] >> >> Wrong. You have been discussing matrices. At least William Hughes >> did. Are you both writing at cross purposes? > > WM is always writing at cross purposes to everyone! I don't take him too seriously as long as it is fun. F. N. -- xyz
From: William Hughes on 18 Nov 2006 15:45
mueckenh(a)rz.fh-augsburg.de wrote: > 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. No. You are trying to show that assuming case iia leads to a contradiction. To do this you need to assume case iia. In particular you are assuming - the set, N, of all natural numbers exists - the set N is infinite. - N has no last element > > > > > 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 nth line is composed of the column indexes 1,2,3,...,n > > > > 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. The diagonal contains all the d_nn. No line contains all the d_nn. Therefore the diagonal is longer than every 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. We have assumed case iia. An infinite set with no last element exists. > > > 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. The set of all lines contains every d_nn. No single line contains every d_nn. The diagonal contains all d_nn. The diagonal is 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. No. The diagonal contains all the d_nn. No line contains all the d_nn. The diagonal is 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. > If we assume case iia the complete column does exist.. > > 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. No. it just shows that the assertion of an actually infinite set of finite numbers leads to results that you do n |