Cantor Confusion
in [Math]
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From: William Hughes on 20 Feb 2007 17:12 On Feb 20, 4:36 pm, mueck...(a)rz.fh-augsburg.de wrote: > On 20 Feb., 16:11, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > You wrote (then snipped) > > > M: [t]he property that every set of even natural numbers must contain > > numbers > > M: larger than its cardinal number, is correct, unless the set > > contains > > M: unnatural numbers. > > > As I noted this is false even in Wolkenmueckenheim. > > You may note what you want. The above statement is true You have said that E does not have a cardinality. You have said that E is a set of even natural numbers. You have said that E does not contain an unnatural number. The statement is not true for E. The statement is not true for "every set" no matter what else may or may not be true. <irrelevent stuff snipped> The statement every set of even natural numbers must contain numbers larger than its cardinal number is not true for potentially infinite sets. So no rephrasing that includes potentially infinite sets can be correct. To make the statment true you must not only state there are no infinite sets, you must also state there are no potentially infinite sets. (of course refusing to call the collection of all even numbers a set will not make it go away). Now for the stuff you snipped. W: Take a property X. Take a potentially infinite set H: A (say the union of all initial segments {1,2,3,...,n}). H: Then, as you note ("The claim in its generality is clearly wrong") H: the statements: H: H: i: Every initial segment {1,2,3,...,n} has H: property X H: H: ii: Every element of A that can be shown to exist H: is a natural number H: H: Do not imply H: H: iii: A has property X. H: H: Sometimes i and ii are true and iii is true. H: Sometimes i and ii are true and iii is false. H: Statements i and ii cannot be used to prove iii. M: They can be used in certain cases. H: Not alone. i and ii are not enough to H: show iii. When someone says H: "iii is false" you reply "but i and ii H: are true". Since i and ii are not enough H: to show iii your replies are empty. Your no comment speaks volumes. - William Hughes
From: Franziska Neugebauer on 20 Feb 2007 17:19 William Hughes wrote: > The statement > > every set of even natural numbers must contain > numbers larger than its cardinal number > > is not true for potentially infinite sets. It it a statement at all? Usually logical statements do not contain modal verbs ("must"). F. N. -- xyz
From: William Hughes on 20 Feb 2007 17:26 On Feb 20, 4:40 pm, mueck...(a)rz.fh-augsburg.de wrote: > On 20 Feb., 16:18, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > On Feb 20, 8:56 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > On 19 Feb., 14:53, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > > In article <1171889781.807587.262...(a)s48g2000cws.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > > > On 18 Feb., 15:53, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > > > > You are right. The claim in its generality is clearly wrong, > > > > > > > So stop using it. Stop claiming > > > > > > > This holds for every initial finite segment therefore > > > > > > it holds for the set. > > > > > > No. Then we must also stop claiming that the set which is the union of > > > > > all initial segments {1,2,3,...,n} contains only natural numbers. > > > > > That is not proven using induction. It follows from the definition of the > > > > union. > > > > like infinity. But it is impossible that both follows, infinity and > > > naturality. > > > Since no one has claimed that infinity is a natural number > > this is not a problem. (Note the claim that "infinity is > > the union of natural numbers" and "infinity is a natural > > number" are two different claims.) > > Note that the number of natural numbers can be mapped one-to-one on > the natural numbers: > > 1,2,3,...,n <---> n i: every initial segment of the natural numbers can be mapped to a natural number. ii: the set of all natural numbers does not contain an element that is not in an initial segment of the natural numbers. iii: The set of all natural numbers can be mapped to a natural number No i: and ii: do not imply iii: > > It is easy to see that natural nunmbers alone do not bring together an > infinite number of numbers. > No i: and ii: do not imply iii: -William Hughes
From: Virgil on 20 Feb 2007 18:25 In article <1172007411.783630.41810(a)a75g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 20 Feb., 16:11, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > You wrote (then snipped) > > > > M: [t]he property that every set of even natural numbers must contain > > numbers > > M: larger than its cardinal number, is correct, unless the set > > contains > > M: unnatural numbers. > > > > As I noted this is false even in Wolkenmueckenheim. > > You may note what you want. The above statement is true if the > statement is true that even in an infinite chain we can conclude from > a<b<c<...<z that a is not larger than z. Since one can at least have an infinite chain of rationals with a < b < c < ... < z in which a < z, there need be no problems with the transitivity of inequality for some sets. Or does WM declare that order relations need not be transistive in Wolkenmueckenheim?
From: Virgil on 20 Feb 2007 18:31
In article <1172007926.277618.98330(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 20 Feb., 20:37, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > On Feb 17, 12:52 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > 0 may be the first (or better the zeroest) ordinal or cardinal number > > > (if you wish to have the empty set in the theory). Nevertheless it is > > > not the first natural number and not a natural number at all. > > > Natural numbers are counting the elements of natural sets, i.e., of > > > sets which exist in reality (in nature, as Cantor woud have said). > > > > So if we call 0 and the positive whole numbers 'mamtural numbers' > > It is enough to call them ordinal numbers. It will not change > mathematics but will help to avoid confusion about the term natural > number. When I say that every set of even natural numbers contains > numbers larger than its cardinality, then I am right with my favourite > meaning of natural numbers but wrong with Bourbaki's. And since Bourbaki is a good deal more knowledgeable about mathematics that WM has any notion of being, let those who are concerned with mathematics by all means follow Bourbaki in all matters in which WM and Bourbaki disagree. |