Cantor Confusion
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From: mueckenh on 13 Nov 2006 08:53 David Marcus schrieb: > > 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. It is difficult to answer this question, because the expression "set" is occupied in modern mathematics by collections of elements which are actually there (you don't know what that means, imagine just a set as you know it). Such infinite sets do not exist. That is the reason why set theory is wrong. But a set which can be understood as growing, could be infinite (if we do not consider the physical restrictions which prevent any infinity). Perhaps you get the idea if you look at the set of all sets. In ZFC it cannot exist. Nevertheless, all sets which exist, exist. So the set of all sets which exist, exists. The question is too difficult to be solved by formalists. > > d. The set of all natural numbers, i.e., {1,2,3,...}, is actually > infinite. No. > > e. An "infinite number" is a number other than the natural numbers. An "infinite number" would be a number other than a natural number. > > f. An actually infinite set must contain an infinite number. If an actually infinite set of numbers existed, and if neighbouring elements had a fixed distance from each other, then the set must contain an infinite number. Regards, WM
From: mueckenh on 13 Nov 2006 09:06 William Hughes schrieb: > 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. > > 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. > > You are confusing cardinals and ordinals. For finite sets they > are the same. For infinite sets they are different. You > want ordinals. The cardinal number aleph_0 is the same as the cardinal number omega. Already Cantor in his later years use omega as a cardinal. > > This cardinal (actually ordinal) number is not an element of the set. May be so with some sets. A set of natural numbers includes its cardinal number. > > > 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. > > BZZZ > > The fact that something is true for all sets of the form > {1,2,3,...n} where n is a finite natural number, > does not mean that it is true for N. Oh yes, exactly that it means, because N consists of nothing else than natural numbers. There are no ghosts in mathematics. > > For example: > For every set B of the form {1,2,3,...,n} there > exists a finite natural number m such that B={1,2,3,...,m}. > However > there is no finite natural number m such that N={1,2,3,...,m} > [This is true even if we assume that N consists of exaclty those > natural numbers that will be named during the lifetime of the > universe]. As long as the "..." denote nothing but natural numbers, your statement is obviously false, as induction proves. If "..." denotes something else, you my be true, but I find that uninteresting. Regards, WM
From: mueckenh on 13 Nov 2006 09:29 Dik T. Winter schrieb: > > Cantor considered well-ordering as a first principle, > > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > > Zermelo was right? > > Cantor did state it without suggesting either that it was a first > principle or something else. He just assumed it. And he was wrong > with that assumption. You are wrong. "Der Begriff der wohlgeordneten Menge weist sich als fundamental für die ganze Mannigfaltigkeitslehre aus. Daß es immer möglich ist, jede wohldefinierte Menge in die Form einer wohlgeordneten Menge zu bringen, auf dieses, wie mir scheint, grundlegende und folgenreiche, durch seine Allgemeingültigkeit besonders merkwürdige Denkgesetz werde ich in einer späteren Abhandlung zurückkommen." (Cantor, Collected works, p.169) > > > His second falsehood is when he states that a set of first cardinality > > > (meaning sets of cardinality aleph-0) can only be counted with use of > > > numbers of the second class (meaning omega and larger). And I think that > > > especially this quote has lead Wolfgang Mueckenheim astray. Also see > > > my discussion about this quote with Dave Seaman. The falsity is apparent > > > if you realise that quote means that every set with cardinality aleph-0 > > > has an omega-th element. > > > > No explicitly stated but implied. > > Explicitly stated. See my discussion with Dave Seaman where I give the > quote, and where there is a long discussion about the meaning. > > > Cantor explained it (as I posted > > recently and repeat it here): > > For the non-German speaking, Cantor explains that > the ordered set has ordinal > 1, 2, 3, ... omega > n+1, n+2, ..., 1, 2, ... n omega+n > 2, 4, 6, ..., 1, 3, 5, ... 2*omega > (Note, in current mathematics the last is noted as omega*2, I will use > the old notation in the sequel.) He is right and indeed, > 1, 2, 4, 8, ..., 3, 6, 12, 24, ..., 5, 10, 20, 40, ... > has ordinality omega^2. A set of cardinality aleph-0 can be ordered > according to every countable ordinality. > > > There is no element omega. But, of course, if the set is to be counted, > > then there must be a number omega following after all natural numbers. > > This is the error. > > No. The above statement is not implied by the letter of Cantor to > Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. > To "count" a set of ordinality omega you do not need omega. In most > cases you need omega, but there is one exception. Can you find it? > To "count" a set of ordinality a with ordinals you need only the > ordinals smaller than a. (And note that in counting with ordinals > you start at 0, because that is the first ordinal.) So you think that counting all natural numbers does not require an infinite number? We agree. > This is a quote from another article by me: > > > I think this has to do with the problems of the conceptions of potential > > > infinity vs. actual infinity that played an important role in mathematics > > > before the early 1900's. Moreso because there were also religious issues. > > > So Cantor crossed the border (after consultation with his religious leader > > > and in strong opposition from him). The actual infinite was impossible > > > (and unreligious) before that time, but he introduced actual infinity. > > > > No. There are hints on the actual infinite in the holy bible and by > > Saint Augustin. Cantor corresponded with Cardinal Franzelin about that. > > The Cardinal did not oppose to the idea. He agreed that God will know > > Cantor's numbers "if they are not contradictory". The Cardinal only > > disagreed with Cantor's "proof" of the actual infinite. Because Cantor > > assumed that God had been forced to create it. > > Perhaps. I do not know the bible, neither what Saint Augustin did write. > It is irrelevant for mathematics. I just was trying to find the reason. Exodus, cap. XV, v. 18: Dominus regnabit in infinitum (aeternum) *et ultra*. Man vgl. die hiermit übereinstimmende Auffassung der ganzen Zahlenreihe als aktual-unendliches Quantum bei S. Augustin (De civitate Dei. lib. XII, cap. 19): Contra eos, qui dicunt ea, quae infinita sunt, nec Dei posse scientia comprehendi. wegen der großen Bedeutung, welche diese Stelle für meinen Standpunkt hat, will ich sie wörtlich hier aufnehmen und behalte mir vor, dieselbe bei einer späteren Gelegenheit ausführlich zu besprechen. ... Energischer, als es hier von S. Augustin geschieht, kann das Transfinitum nicht verlangt, volkommener nicht begründet und verteidigt werden. > > And, I think that Cantor would agree that > 1, 3, 4, ..., 2 > also has the number omega + 1. Yes, I think so too. Regards, WM
From: mueckenh on 13 Nov 2006 09:37 William Hughes schrieb: > > Every column has a fixed number omega of terms. You can add 1 further > > term and you will get omega + 1 terms in every column. Every line has > > less than omega terms. If you add 1 term to every line, the number of > > terms in every line remains less than omega. What about the diagonal? > > It has to satisfy both conditions, which is impossible. > > No. Adding one element to each line does not change > the supremum of the lengths of the lines, so it does > not change the length of the diagonal. The supremum is larger than any line, so it must be larger han the diagonal which is confined by the lines. > > The number of terms in each line, n, is less than omega. > The number of columns is the supremum of the number of terms > in each line. The number of columns is omega. If there is no line with omega terms, what does the supremum of the columns consists of? > The number of lines is omega. > The number of columns is equal to the number of lines. You can repeat your prayer as often as you like. It will not be heard. > > > Now add one term to each line. > > The number of terms in each line, n+1, is less than omega. > The number of columns is the supremum of the number of terms > in each line. The number of columns is omega. > The number of lines is omega. > The number of columns is equal to the number of lines. You can repeat your prayer as often as you like. It will not be heard. Regards, WM
From: mueckenh on 13 Nov 2006 09:42
William Hughes schrieb: > > Every column has a fixed number omega of terms. You can add 1 further > > term and you will get omega + 1 terms in every column. Every line has > > less than omega terms. If you add 1 term to every line, the number of > > terms in every line remains less than omega. What about the diagonal? > > It has to satisfy both conditions, which is impossible. > > No. Adding one element to each line does not change > the supremum of the lengths of the lines, so it does > not change the length of the diagonal. > > The number of terms in each line, n, is less than omega. > The number of columns is the supremum of the number of terms > in each line. The number of columns is omega. > The number of lines is omega. > The number of columns is equal to the number of lines. > > > Now add one term to each line. > > The number of terms in each line, n+1, is less than omega. > The number of columns is the supremum of the number of terms > in each line. The number of columns is omega. > The number of lines is omega. > The number of columns is equal to the number of lines. PS: You forgot to consider the case that one term is added to each column. The ordinal of the columns is then larger than omega. Regards, WM |