Cantor Confusion
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From: Franziska Neugebauer on 13 Nov 2006 02:35 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. F. N. -- xyz
From: Dik T. Winter on 13 Nov 2006 07:20 In article <1163351315.961078.253970(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > What matters is not which ordinals you use > > to occupy places, but where you put the "..." > > For example. The both sequences > > > > 5,4,3,2,1,6,7,8 ... > > omega, omega*omega, omega+5*omega^3 + 7, 1,2.3, ... > > > > represent the ordinal omega. > > > The first is omega. The second is not omega. You see it if you replace > omega by 1,2,3,... Perhaps you wanted to say that they both represent > the *cardinal* omega. That is correct. If you want to work in the model were each ordinal number is the ordered set of all its predecessors, than also you can not replace omega by 1, 2, 3, ... You have to replace it by {1, 2, 3, ...}. So the sequence omega, 1 is equivalent to the sequence: {1, 2, 3, ...}, 1 and has two elements. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 13 Nov 2006 07:28 In article <1163352469.769509.39380(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > 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. Again a slight confusion between ordinal numbers and natural numbers. 0 is also an ordinal number. So each ordinal counts the number of preceding ordinals: |{0, 1, 2}| = 3 |{0, 1, 2, 3, ...}| = omega But obviously: |{0, 1, 2, 3, ...}| = |{1, 2, 3, ...}| because there is an order invariant bijection between the two ordered sets. > 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. Wrong. When we look at initial segments of ordinals we *need* to start at 0 because that is also an ordinal. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 13 Nov 2006 07:37 In article <J8o4xt.J8C(a)cwi.nl> I wrote some thing wrong. Here a corrected version: In article <1163351315.961078.253970(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > What matters is not which ordinals you use > > to occupy places, but where you put the "..." > > For example. The both sequences > > > > 5,4,3,2,1,6,7,8 ... > > omega, omega*omega, omega+5*omega^3 + 7, 1,2.3, ... > > > > represent the ordinal omega. > > > The first is omega. The second is not omega. You see it if you replace > omega by 1,2,3,... Perhaps you wanted to say that they both represent > the *cardinal* omega. That is correct. If you want to work in the model were each ordinal number is the ordered set of all its predecessors, than also you can not replace omega by 1, 2, 3, ... You have to replace it by {0, 1, 2, 3, ...}. So the sequence omega, 1 is equivalent to the sequence: {0, 1, 2, 3, ...}, 1 and has two elements. (Note that 0 is also an ordinal.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 13 Nov 2006 08:49
David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > > > > You carried out a survey? Or is that your guess? I know some > > > > mathematicians who know what a mathematician should understand by > > > > "limit", "number", "grow". > > > > > > Please post their definitions of the words. > > > > Try to get a clue of Cantor's second creation principle. Then you will > > know these things. > > I suppose that means you are unable to define them. Not with words you know, I suppose. Bu if you also are unable to understand Cantor or others, explainining his principle, then you should see that its your turn to learn some words. > > > > > They were discussing mathematics. Philosophy belongs to mathematics. > > > > > > That's nonsense. > > > > That's ignorance. > > If your "that's" refers to the same thing as my "that's", then I agree. It refers to the same thing as your "that's" is, it does not refer to the same thing as your "that's" refers to. > > > Several reasons: They were discussing philosophy of mathematics, not > > > mathematics itself. You don't understand what they were saying. > > > > They said: "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." > > Indeed. If people *object* to an axiom, that is philosophy. But if people choose a set of axioms, that is what? >Everyone is > welcome to choose their own axioms. That's mathematics? > > > > And, you haven't given a definition of the term. > > > > I thought you'd know transitivity: Take the expression "finshed entity" > > where entity is a variable like "the set X" in set theory. Now replace > > this variable by a fixed set like N, which in mathematics, is an > > infinite process. This leads to "finished infinite process", > > abbreviated by "finished infinity". Was this simple enough for you to > > understand? > > Amusing, but that just shows how you can make up new terms. You still > haven't provided a defintion. It appears you are saying that "finished > infinite process" and "finished infinty" are synonyms. Fine. But, you > haven't defined either one. Please define them. Would like to do. Please le me know which words are available in your universe of discourse. > > > > > Which mathematics do you allude to? > > > > > > That which is taught in school, explained in textbooks, published in > > > journals, and discussed by some in this newsgroup. > > > > I teach at a school, I wrote textbooks (and my new book is in print), I > > published in journals and I wrote in newsgroups. > > Irrelevant since that isn't what my sentence means in English. Could you explain what your sentence means? For instance: How many schools and textbooks do you require? Regards, WM |