Cantor Confusion
in [Math]
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From: Dik T. Winter on 5 Mar 2007 09:08 In article <1173083261.665454.244340(a)s48g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > 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. .... > > > 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. The introduction to chapter 3 tells us: "The main result of the preceding section is a characterization of the real line as the unique linear ordering without endpoints that is complete and has a countable subset dense in it. This is the usual departure point for the study of toplogical properties of the real line. We give some basic definitions and theorems of the subject in this section. Our objectives are to justify the claim that set theory, as we developed it so far, provides a satisfactory foundation for analysis ..." So they state, quite explicitly, that the theorems and definitions in that section actually do *not* belong to set theory. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Mar 2007 09:55 In article <JEFoLu.6x1(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <1173083261.665454.244340(a)s48g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > > Karel Hrbacek and Thomas Jech: "Introduction to Set Theory" Marcel > > > > Dekker Inc., New York, 1984, 2nd edition. .... > > > No. Definitions of limits are beyond set theory. > > > > Today nearly everything in mathematics is set theory. These limits > > above are set theory. > > The introduction to chapter 3 tells us: > "The main result of the preceding section is a characterization of the real > line as the unique linear ordering without endpoints that is complete and > has a countable subset dense in it. This is the usual departure point for > the study of toplogical properties of the real line. We give some basic > definitions and theorems of the subject in this section. Our objectives > are to justify the claim that set theory, as we developed it so far, > provides a satisfactory foundation for analysis ..." > So they state, quite explicitly, that the theorems and definitions in that > section actually do *not* belong to set theory. To be honest, they *do* define a set theoretic limit. Chapter 10, section 2, page 193. Limits of transfinite sequences. And by *that* definition: lim{n -> omega} = omega. This limit definition is similar to the limit definition I gave earlier. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 5 Mar 2007 10:46 In article <1173083672.999163.257670(a)s48g2000cws.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 5 Mrz., 03:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > 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. As a set of nodes, it indeed does have cardinality aleph_0. > Result: The union tree U(T(n)) is too narrow to contain more than > aleph_0 paths. Wrong. Each path is a set of nodes, hence a subset of the complete set of nodes, hence an element of the powerset of this set of nodes. We know that this powerset is not countable, so the set of paths can be (and actually is) uncountable. -- 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 6 Mar 2007 05:27 On 5 Mrz., 16:46, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1173083672.999163.257...(a)s48g2000cws.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 5 Mrz., 03:14, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > 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. > > As a set of nodes, it indeed does have cardinality aleph_0. Of course. And we need not use the limit definition (of Hrbacek and Jech or else) in order to get this result, because the whole tree T(oo) has only a countable set of nodes. Therefore C(oo) cannot be larger than countable. > > > Result: The union tree U(T(n)) is too narrow to contain more than > > aleph_0 paths. > > Wrong. Each path is a set of nodes, hence a subset of the complete > set of nodes, hence an element of the powerset of this set of nodes. > We know that this powerset is not countable, so the set of paths can > be (and actually is) uncountable. This argument should be recognizable as invalid. It could only be applied if ALL combinations of nodes were paths. But this is not the case! Not every set of nodes is a path, as you well know. In order to count the paths (-bundles) which pass through a certain level of the tree, it is sufficient to know the cross section of the tree. More than C(n) paths (-bundles) cannot fit into the tree. This proves without any doubt that the complete tree T(oo) cannot contain more than countably many paths (-bundles). But we know that T(oo) contains all existing real numbers of [0, 1] represented as paths. Regards, WM
From: Dik T. Winter on 6 Mar 2007 09:12
In article <1173176849.548735.27970(a)n33g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 5 Mrz., 16:46, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > 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. > > > > As a set of nodes, it indeed does have cardinality aleph_0. > > Of course. And we need not use the limit definition (of Hrbacek and > Jech or else) in order to get this result, because the whole tree > T(oo) has only a countable set of nodes. Therefore C(oo) cannot be > larger than countable. The latter is an invalid conclusion. > > Wrong. Each path is a set of nodes, hence a subset of the complete > > set of nodes, hence an element of the powerset of this set of nodes. > > We know that this powerset is not countable, so the set of paths can > > be (and actually is) uncountable. > > This argument should be recognizable as invalid. What is invalid? I may note that the parenthetical remark is *not* part of the argument. > It could only be > applied if ALL combinations of nodes were paths. But this is not the > case! Not every set of nodes is a path, as you well know. In order to > count the paths (-bundles) which pass through a certain level of the > tree, it is sufficient to know the cross section of the tree. More > than C(n) paths (-bundles) cannot fit into the tree. This proves > without any doubt that the complete tree T(oo) cannot contain more > than countably many paths (-bundles). This proves that in no way. > But we know that T(oo) contains > all existing real numbers of [0, 1] represented as paths. Indeed. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |