Cantor Confusion
in [Math]
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From: Dik T. Winter on 4 Mar 2007 21:14 In article <1172998862.730699.246980(a)n33g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 4 Mrz., 02:55, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > I missed something. What is the definition of the cross section of > > > > the "unit tree"? What is the definition of the "unit tree"? > > > > > > What is the definition of the last parentheses of the proof of the > > > divergence of the harmonic series? > > > > There is no such definition because there is no last parenthesis. If > > there were a last parenthesis the harmonic series would converge. > > Nevertheless you know the harmonic series and can address every pair > of parentheses and know the number of terms in the whole series. Indeed. But there is no last pair. > > What is the "cross section" of the union tree? > > I said already several times: The cross section of a tree is the > number of nodes in its last level. > In order to get the cross section C(oo) of the infinite tree, > enumerate the nodes of the levels of T(oo) by > {1} > {1,2} > {1,2,3,4} > ... > take the union of all of these sets, and take the cardinal number of > this union. Nonsense. The cross section of a tree is the number of nodes in its last level. So you now claim that the number of nodes in its last level of T(oo) is aleph-0? But there is no last level. -- 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 5 Mar 2007 03:27 On 5 Mrz., 03:08, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172917355.563795.27...(a)n33g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 3 Mrz., 02:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1172774506.011332.315...(a)p10g2000cwp.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > > Every nondecreasing sequence of real numbers bounded from above has a > > > > limit. There is a limit superior and a limit inferior proven by set > > > > theory. > > > > > > Not defined by set theory. > > > > Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" Marcel > > Dekker Inc., New York, 1984, 2nd edition. > > > > 3.8 Theorem Every bounded sequence of real numbers has a > > convergent subsequence. > > You are doing it again. Throwing around Microsoft specific codings that > have not been defined elsewhere. I have great problems reading this stuff. > I will try to translate it: Sorry, but the exact meaning of these sentences is not of interest. It is simple stuff from first semester. This quote should only show you that "limit" is a word frequently occuring in a book on set theory. > > > The number a = sup {inf {ak | k <= n} | n in N} used in the proof of > > Theorem 3.8 is called the limit inferior of 'an'. Similarly b = inf > > {sup {ak | k <= n} | n in N} is called the limit superior of 'an'. Prove > > that (a) b exists for every bounded sequence and a < b. > > > > Of course definitions and proofs are done by set theory > > No. Definitions of limits are beyond set theory. Today nearly everything in mathematics is set theory. These limits above are set theory. Regards, WM
From: mueckenh on 5 Mar 2007 03:34 On 5 Mrz., 03:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172998862.730699.246...(a)n33g2000cwc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 4 Mrz., 02:55, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > > I missed something. What is the definition of the cross section of > > > > > the "unit tree"? What is the definition of the "unit tree"? > > > > > > > > What is the definition of the last parentheses of the proof of the > > > > divergence of the harmonic series? > > > > > > There is no such definition because there is no last parenthesis. If > > > there were a last parenthesis the harmonic series would converge. > > > > Nevertheless you know the harmonic series and can address every pair > > of parentheses and know the number of terms in the whole series. > > Indeed. But there is no last pair. > > > > What is the "cross section" of the union tree? > > > > I said already several times: The cross section of a tree is the > > number of nodes in its last level. > > In order to get the cross section C(oo) of the infinite tree, > > enumerate the nodes of the levels of T(oo) by > > {1} > > {1,2} > > {1,2,3,4} > > ... > > take the union of all of these sets, and take the cardinal number of > > this union. > > Nonsense. The cross section of a tree is the number of nodes in its > last level. So you now claim that the number of nodes in its last level > of T(oo) is aleph-0? But there is no last level. Nevertheless, all last levels are there which make up the set of all last levels of finite trees. If the set of all natural numbers exists (e.g., as the set of all last elements n of initial segments {1,2,3,...,n}), then the set of all last levels exists too. And we know that this set of all last levels does not contain any element which, as a set of nodes, has a cardinality of more than aleph_0. Result: The union tree U(T(n)) is too narrow to contain more than aleph_0 paths. Regards, WM
From: Virgil on 5 Mar 2007 03:43 In article <1173083261.665454.244340(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 5 Mrz., 03:08, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172917355.563795.27...(a)n33g2000cwc.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > > 3.8 Theorem Every bounded sequence of real numbers has a > > > convergent subsequence. > > > > You are doing it again. Throwing around Microsoft specific codings that > > have not been defined elsewhere. I have great problems reading this stuff. > > I will try to translate it: > > Sorry, but the exact meaning of these sentences is not of interest. It > is simple stuff from first semester. This quote should only show you > that "limit" is a word frequently occuring in a book on set theory. First of all, exact meanings are of critical importance in every part of mathematics, including set theory, and anyone, like WM, who says otherwise is mathematically incompetent. The word "limit" only occurs formally in math texts after being properly defined, and none of those proper mathematical definitions are compatible with WM's incompetently vague notions of limits.
From: Virgil on 5 Mar 2007 03:51
In article <1173083672.999163.257670(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 5 Mrz., 03:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172998862.730699.246...(a)n33g2000cwc.googlegroups.com> > > > > Nonsense. The cross section of a tree is the number of nodes in its > > last level. So you now claim that the number of nodes in its last level > > of T(oo) is aleph-0? But there is no last level. > > Nevertheless, all last levels are there which make up the set of all > last levels of finite trees. A set of all last levels does not, itself, have to have a last level, just as the set of /all/ naturals cannot, at least in any sane set theory, contain a last member. > If the set of all natural numbers exists > (e.g., as the set of all last elements n of initial segments > {1,2,3,...,n}), then the set of all last levels exists too. No one objects to its existence, so long as no one claims that it must have a last level of its own. If is WM's repeated insistence that a set which cannot contain a last member must contain a last member which is the ongoing problem. > And we > know that this set of all last levels does not contain any element > which, as a set of nodes, has a cardinality of more than aleph_0. Then that proves that it that union cannot contain a last level, such such a "last level" would have to be a complete infinite binary tree, and thus could not be countable. > > Result: The union tree U(T(n)) is too narrow to contain more than > aleph_0 paths. And too narrow to contain a complete infinite binary tree, so it must be incomplete. |