Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Dik T. Winter on 13 Nov 2006 22:08 In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > So lets count the set of all natural numbers {1,2,3,...} > There are no natural number left. So we stop > using natural numbers and use ordinals > (and to nobody's surprise a few things change). This is wrong. There is no ordinal needed to count the elements of the set of all natural numbers. You can count until you weigh an ounce (;-)) but you will never finish. Neither the elements you wish to count will be exhausted nor the numbers with which you count. I think this is potential infinity. On the other hand, when you ask "how many" elements there are in N, you need an infinity (and this is, I think, actual infinity). But all this hinges quite closely on the semantics of the word "count". If seen as a process, you do not need an infinity; when seen as the result of a process, you do need an infinity. In many languages (German and Dutch amongst others) there are different words for the two meanings, but the meanings are conflated in English. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 13 Nov 2006 22:40 In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com>, "William Hughes" <wpihughes(a)hotmail.com> wrote: > The set {12,3,...} has omega elements, > and the set {1,2,3,...,omega} has omega + 1 elements, and > the set {1,2,3,...,omega,omege+1} has omega + 2 elements etc. > Now for a set of the form {1,2,3, ... , a} the ordinal of the > set is a+1. This is because sets of ordinals do not act > quite like sets of natural numbers. > > - William Hughes One has to be a bit careful with this sort of problem to distinguish between ordinal and cardinal numbers. To have the same ordinal number requires an order isomorphism preserving all the the order properties as well as being a bijection, whereas having the same cardinality only requires bijection. The set {1,2,3,...} has ordinal number omega and cardinal number aleph_0. The set {1,2,3,...,omega} has ordinal omega + 1 and cardinal aleph_0, {1,2,3,...,omega,omega + 1} has ordinal omega + 2 and cardinal aleph_0. And so on.
From: William Hughes on 13 Nov 2006 23:56 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > omega 1,2,3,4... > > > > omega+1 2,3,4 ...,1 > > > > omega+2 3,4,5 ...,1,2 > > > > 2*omega 1,3,5... 2,4,6... > > omega = 1,3,5,... > omega = 2,4,6,... Ok, if we read the "=" is the ordinal of the sequence. > omega, omega = 2omega No You want to do is something like take 2omega is the ordinal of the sequence 1,3,5,...,2,4,6 ... and then substiute omega. But what you are saying is the ordinal of the sequence 1,3,5,...,2,4,6 ... is the same as the ordinal of the sequence omega,omega, which is nonsense. The sequence omega,omega has an ordinal of 2. > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 This is true (it is of course not the only sequence that represents the ordinal 2omega +1) However, even using your nonsensical formal manipulations we get 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 (1,3,5,...),(omega,2,4,6,...),omega = 2omega +1 omega,omega,omega = 2omega+1 Which does not contradict anything you wrote. Which nonsensical formal manipulation were you thinking of? > > > > It can however be a term in another sequence representing > > omega+1. There is no single sequence which represents > > omega+1. > > > > > but omega is the first part of it, omega = 2,3,4,... > > > > > > This dilemma is what I have been trying to explain for years now. > > > > > > > The explanation is that if we do not insist that an ordinal be > > represented by an initial seqment of ordinals, then there > > is no unique representation of an ordinal. > > I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, > ... > omega + 1 = 1,2,3,..., omega. > > Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. No. No intial segment of natural number that has a maximum can be counted by an unnatural number like omega. The set of natural numbers {1,2,3,...} does not have a maximum. >This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Obviously something has > been lost. > No, taking the ordinal number of a set of natural numbers is not the quite the same as taking the ordinal number of a set of ordinals (natural numbers start at 1, ordinals start at 0). For any initial sequence of ordinals with a maximum, |{0,1,2,...,a}| = a+1 regardless of the kind of a. This holds for a in N and for a>= omega. So clearly nothing has been lost. - William Hughes
From: mueckenh on 14 Nov 2006 06:24 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > Franziska Neugebauer schrieb: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > Franziska Neugebauer schrieb: > >> >> [...] > >> >> >> Are there really three vertices in WM's "triangle"? > >> >> > If finished infinities [...] > >> >> Verbiage. > >> > Yes. But, sorry to see, it is the fundament of modern mathematics. > >> "Finished infinities" is your wording. > > Precisely describing the fundament of modern mathematics. > > Cbjre Bs Oryvrs. Always insisting on having the last word? Regards, WM
From: mueckenh on 14 Nov 2006 06:32
Virgil schrieb: > In article <1163425753.113699.136190(a)b28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > >Everyone is > > > welcome to choose their own axioms. > > > > That's mathematics? > > If one chooses a set of axioms no one else is interested in it is pretty > much useless mathematics, but still mathematics. But to reject an axiom is philosophy? > > > Any definition of a "finished" anything which is based on the word > "finished" itself, is circular at best. I did not intend to define it but explain that this expression is used in current mathematics. > Do you claim to teach mathematics at your school? If so, at what level > and with what content. No, I do not claim it, I simply do it. Analysis I + II, Algebra I + II, for first and second semester. History of mathematics, and history of the infinite for students of all semesters. > If you do not, then whatever you do claim to teach is irrelevant. > > Are any of your textbooks primarily mathematics texts? If not, what is > their primary subject matter, not that it does matter. Primarily mathematical textbooks: Algebra und Geometrie Differential- und Integralrechnung Kleine Geschichte der Mathematik Die Geschichte des Unendlichen And see my new book to appear this year: Die Mathematik des Unendlichen (? 28.50) Regards, WM |