Cantor Confusion
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From: David Marcus on 11 Nov 2006 17:11 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > So let's be definite here. Do you have a strong belief that your > > argument can be expressed in the language of set theory and uses only > > first order logic applied to the axioms of set theory? > > I need no belief. If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. The most dubious assumption in that statement is that your argument is "consistent mathematics". -- David Marcus
From: mueckenh on 11 Nov 2006 17:14 Dik T. Winter schrieb: > > In mathematics counting and comparing numbers comes before most other > > things. Not because I think so, but because mathematics developed this > > way. > > In that case you are talking about a different kind of mathematics than > what I am talking about. Yes, I am talking about the mathematics which was developed over the last 500 years. > > > > Where are they? That there are "limit ordinals" does not mean there are > > > "limits" in set theory. > > > > Why are they called so? > > Ask the people who coined those names. Cantor called them "Grenzzahlen" and denoted them by Lim_n etc. Because by his second creation principle these numbers are created by limit processes. "wir nennen sie Zahlen zweiter Art, sind so beschaffen, daÃ? es für sie eine nächstkleinereï? gar nicht gibt; diese gehen aber aus Fundamentalreihen als deren Grenzzahlen hervor" > > > > Since Cantor quite a bit has changed. Limits are *not* part of set > > > theory, they belong to topology and other things build on set theory. > > > > Experience has shown that practically all notions used in contemporary > > mathematics can be defined, and their mathematical properties derived, > > in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not > > include limits? > > Darn. Do read. It does include limits, but not in the branch of set > theory. All "modern mathematics" is set theory. Set theory is not a branch but the foundation, in fact, it is all. > In that branch the term has not been defined, and so is > meaningless. If you want to use such a term in set theory, you have > to define it in terms of set theory. Cantor did. Should it meanwhile have been forgotten or abolished? "a = Lim_n (a_n) a ist hier die auf sämtliche Zahlen a_n der GröÃ?e nach nächstfolgende Zahl." Regards, WM
From: mueckenh on 11 Nov 2006 17:16 Franziska Neugebauer schrieb: > WM does not belong to the "factinista". He is ignorant about the fact > that nowadays mathematics is a formal science in the first place. The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works. (Bertrand Russell) Die Mathematik ist wie die Dialektik ein Organ des höheren Sinns. In der Ausübung ist sie eine Kunst wie die Beredsamkeit. Für beide hat nichts Wert als die Form; der Gehalt ist ihnen gleichgültig. (Johann Wolfgang von Goethe) Mathematik kann nie durch Logik allein begründet werden. Als Vorbedingung zur Anwendung logischer Schlüsse ist uns bereits immer etwas gegeben: gewisse außerlogische konkrete Objekte, die anschaulich als unmittelbares Erlebnis vor dem Denken da sind. (David Hilbert) Der Lebensnerv der mathematischen Wissenschaft ist bedroht durch die Behauptung, Mathematik sei nichts anderes als ein System von Schlüssen aus Definitionen und Annahmen, die zwar in sich widerspruchsfrei sein müssen, sonst aber von der Willkür des Mathematikers geschaffen werden. Wäre das wahr, dann würde die Mathematik keinen intelligenten Menschen anziehen. Sie wäre eine Spielerei mit Definitionen, Regeln und Syllogismen ohne Ziele und Sinn. Die Vorstellung, daß der Verstand sinnvolle Systeme von Postulaten frei erschaffen könne, ist eine trügerische Halbwahrheit. (R. Courant, H. Robbins) Aber ist denn das eine Wissenschaft, die Sätze beweist, ohne zu wissen, was sie beweist? (G. Frege) Gruß, WM
From: David Marcus on 11 Nov 2006 17:23 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. > > > > 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. > > > > > > 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. > > > > So, it remains true that "finished > > > > infinity" does not have a mathematical meaning. > > Ironically you are even right, for once, though you don't know it. Notice the quotes. You really should learn to read more carefully. > > > This term has been used in a mathematical discussion by Hrbacek and > > > Jech. I don't see why I shouldn't do the same. > > > > 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. Everyone is welcome to choose their own axioms. > > 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. > > > 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. -- David Marcus
From: David Marcus on 11 Nov 2006 17:32
mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > William Hughes wrote: > > > > > a: Any infinite set of numbers must contain an infinite number > > > > > > b: It is possible to have an infinite set of numbers that does > > > not contain an infinite number. > > > > WM, please tell us if you agree or disagree with the statements a and b > > above. > > You will not understand it. But I'll try. > > The potentially infinite set N does not contain an infinite number. In > this case (b) is correct. > > An actually infinite set is a set which has a cardinal number like > omega. In set theory every set is actually infinite. So N is actually > infinite. If its cardinal number omega is claimed to count the members > of N, then there is a contradiction, because omega counts all n in N > but does not belong to N while the initial segments of N are counted by > natural numbers. This statement remains true as far as the natural > numbers reach, i.e., for all natural numbers , i.e., for N. > > The contradiction becomes obvious in the matrix discussed or in the > following few lines: > > Cantors first approach read: n+1, n+2, n+3, ..., 1, 2, 3, ..., n is a > sequence with ordinal number omega + n. > Here omega does not appear in the sequence. omega is nothing but the > sequence > n+1, n+2, n+3, .... But Cantor wanted omega to be an ordinal number > alike the finite ordinals. Therefore, omega must appear in the > sequence: > n+1, n+2, n+3, ..., omega, 1, 2, 3, ..., n > What is the ordinal number of this sequence? It appears that you are saying that the terms "infinite set" and "infinite number" are not meaningful. There are only "potentially infinite sets" and "actually infinite sets". Is that correct? Do you agree with the following statements? c. It is possible to have a potentially infinite set of numbers that does not contain an infinite number. d. The set of all natural numbers, i.e., {1,2,3,...}, is actually infinite. e. An "infinite number" is a number other than the natural numbers. f. An actually infinite set must contain an infinite number. -- David Marcus |