Cantor Confusion
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From: Dik T. Winter on 15 Nov 2006 20:42 In article <1163606442.944751.259900(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > Cantor considered as "Anzahl" which can be determind by "abzaehlen" > > > simply the ordinal number of a set. > > > > But in that quote there is no mention of "Anzahl". There is only mention > > of "abz?hlen", and if I understand German well, that means, in general, > > just like in Dutch, the "process of counting". > > That is right. But Cantor has a slightly different understanding. That means that he understands under "abz?hlen" something different? > He > defines: Die kleinste M?chtigkeit, welche ?berhaupt an unendlichen, > d. h. aus unendlich vielen Elementen bestehenden Mengen auftreten kann, > ist die M?chtigkeit der positiven ganzen rationalen Zahlenreihe; ich > habe die Mannigfaltigkeiten dieser Klasse ins unendliche abz?hlbare > Mengen oder k?rzer und einfacher abz?hlbare Mengen genannt; sie sind > dadurch charakterisiert, da? sie sich (auf viele Weisen) in der Form > einer einfach unendlichen, gesetzm??igen Reihe ... darstellen lassen, > so da? jedes Element der Menge an einer bestimmten Stelle dieser Reihe > steht und auch die Reihe keine anderen Glieder enth?lt als Elemente > der Menge. (Collected Works, p. 152) Apparently no. What he is describing here is bijections. And what he is stating is that each countable set can be put in bijection with another countable set. But that is about cardinals, not about ordinals. Sets with the same "M?chtigkeit" can be put in bijection with each other (this is a generalisation of the above statement). However, the process of counting ("abz?hlen") requires an order preserving bijection. Most ordered sets of cardinality aleph-0 require omega to get such an order preserving bijection with the ordinals. But there is one exception, namely the ordered set with ordinality omega. > > There is one set of the > > first cardinality (N) that can be counted (i.e. the process of counting) > > without any reference to w. That is potential infinity, and you did > > agree. And that is still the case in modern set theory. > > Cantor would say: "It can be counted into the infinite", later he > dropped the specification "into the infinite". Perhaps. > > > This does *not* mean that omega is an element of the set. > > > > The quote means that during the process of "abz?hlen" we need a number > > of the second class, and so there is an omega-th element (not an > > element omega). > > I cannot agree. With what? (Strange enough, in other articles you state that the set N does contain an un-natural number...) > > > {1,2,3} is the collection of, and a convenient expression to write that > > > we are talking about, the numbers 1 ,2, and 3. > > > > No, the set *containing* the numbers 1, 2 and 3. > > The envelope becomes more important than the contents. A characteristic > of modern times. The publisher becomes more important than the author. > The director becomes more important than the composer. The trainer > becomes more important than the football-team. Rethoric. > > > > Sorry, I do not understand what you write here. The set > > > > {{1, 2}, {3, 4}} > > > > has two elements. It's cardinality is 2. And each of the elements > > > > is a set containing two elements. > > > > > > Yes, here you are talking about two unordered pairs of numbers. You use > > > a convenient way to denote that. Nothing else stands behind the {, } > > > symbols. In particular N = 1,2,3,... = {1,2,3,...}. That does not > > > prevent to build a set {{1,2,3,...}, 1} with two elements. > > > > And omega = {0, 1, 2, ...} (I use { and } here to denote ordered sets.) > > That is not customary. An ordered set is more than a set, because the > order is added. Yes, that is why I state that I use it as such. > > So you can not replace omega by 0, 1, 2, ... > > Well, that is correct. No, you replaced it by 1, 2, 3, ...; but that is also not allowed. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 16 Nov 2006 01:27 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > 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? Of course. > > > > 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. I told you several times that the terminology in any modern textbook is fine. For some reason you do not like this answer. -- David Marcus
From: David Marcus on 16 Nov 2006 01:32 mueckenh(a)rz.fh-augsburg.de wrote: > 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. The words in my question were supposed to be as you interpret them, not as they are used in ZFC or standard mathematics. Since you can't/won't define your terms, perhaps we can figure out what you are talking about by asking some questions. > > d. The set of all natural numbers, i.e., {1,2,3,...}, is actually > > infinite. > > No. What would be something that is "actually infinite"? > > e. An "infinite number" is a number other than the natural numbers. > > An "infinite number" would be a number other than a natural number. Are you agreeing or disagreeing? > > 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. Is that a "no" or a "yes"? -- David Marcus
From: David Marcus on 16 Nov 2006 01:35 Virgil wrote: > In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > 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. > > While infinite collections in any physical sense are not possible, why > are imaginary infinities, such as sets of numbers must be, unimaginable? For that matter, we can always switch from Platonism to formalism and declare the question of whether sets really exist to be a philosophical question. -- David Marcus
From: David Marcus on 16 Nov 2006 01:43
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > 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. How do you know this? Do you have any sort of rationale or proof? It seems such a silly thing to say. Consider: Each of the following sequences has a last element: 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 .... This sequence does not have a last element: 1 2 3 4 5 ... This last sequence has three dots on the right. None of the other sequences do. So, this last sequence is clearly different in some way from all the other sequences. -- David Marcus |