Cantor Confusion
in [Math]
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From: Virgil on 3 Mar 2007 15:32 In article <1172919142.768602.212630(a)z35g2000cwz.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 3 Mrz., 02:18, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172835359.498019.275...(a)64g2000cwx.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > The cross section of the tree is the same as the number of unit > > > fractions in the last parenthesis used by Oresme. > > > > 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? What is the definition of the last natural number? Does WM claim that until it can be defined, natural numbers cannot exist? > It is not defined, but all > parentheses are required to exist. Just like the naturals. > > > > Proof please that that cross section is well ordered. Anyhow, a cross > > > > section that you refuse to define? > > > > > > C(oo) =< Card(U(T(n))) > > > > So you refuse to define it? As long as you do not define it, it can not be > > ascertained whether the statement above is true or false. > > The union of all trees is defined in your opinion? The "union" of s set of trees is NOT defined when one uses the standard definition of a tree. > The cross section C(oo) can be defined as the limit of all finite > cross sections. Define this sort of limit. > It can be defined as the cross section of the union tree U(T(n)). Which union is not defined. > It can also be defined as the cross section of the of the complete > tree T(oo). The only complete infinite binary tree is quite different from WM's version of U({T(n): n e N}). That complete tree demonstrably has uncountably many paths. > > If we use the definition of C(oo) as the cross section of the union > tree U(T(n)), then we have the following facts: > > The tree T(n) can mapped on the natural number of its nodes. > > There are infinitely many trees in the union U(Tn)) as there are > infinitely many natural umbers in the union N. > The union has a finite size concerning the sizes of all natural > numbers. None of them is infinite. Therefore the nodes of the union > tree have a finite cardnality. (Recall every union of finite trees is > the largest tree in the union.) Except for every presumed "largest tree" there is a larger tree, so that WM's delusion fails. > > If we use the definition of C(oo) as the cross section of T(oo) then > we have the estimation: The cross section cannot surpass the number of > all nodes. This number is coutable, hence C(oo) =< Card(T(oo)) = > aleph_0. Then, by all means, let us avoid any definitions which lead to such self contradictory conclusions.
From: Dik T. Winter on 3 Mar 2007 20:55 In article <1172919142.768602.212630(a)z35g2000cwz.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 3 Mrz., 02:18, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172835359.498019.275...(a)64g2000cwx.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: .... > > > The cross section of the tree is the same as the number of unit > > > fractions in the last parenthesis used by Oresme. > > > > 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. > It is not defined, but all > parentheses are required to exist. It is not defined because the last one does not exist. > The same is valid for the cross > sections of the trees. So you refuse to give a definition. > > > > > The set of unions cannot be larger than the cross section of the > > > > > tree U(T(n)) which is Card(2^omega). > > > > > > > > Proof please that that cross section is well ordered. Anyhow, a cross > > > > section that you refuse to define? > > > > > > C(oo) =< Card(U(T(n))) > > > > So you refuse to define it? As long as you do not define it, it can not be > > ascertained whether the statement above is true or false. > > The union of all trees is defined in your opinion? Yes. > The cross section C(oo) can be defined as the limit of all finite > cross sections. There is no such limit, unless you define such a limit. > It can be defined as the cross section of the union tree U(T(n)). That is not a definition, because (according to your definition of cross section), U(T(n)) has no cross section. > It can also be defined as the cross section of the of the complete > tree T(oo). Again a non-definition. > If we use the definition of C(oo) as the cross section of the union > tree U(T(n)), then we have the following facts: What is the "cross section" of the union tree? > The tree T(n) can mapped on the natural number of its nodes. > > There are infinitely many trees in the union U(Tn)) as there are > infinitely many natural umbers in the union N. > The union has a finite size concerning the sizes of all natural > numbers. None of them is infinite. Therefore the nodes of the union > tree have a finite cardnality. (Recall every union of finite trees is > the largest tree in the union.) There is no logic in this paragraph, much less mathematics: > If we use the definition of C(oo) as the cross section of T(oo) then > we have the estimation: How do you *define* that cross section? According to your definition, the cross section of a tree is the set of nodes at the last level. As there is no last level in T(oo), we can not use that definition. > The cross section cannot surpass the number of > all nodes. This number is coutable, hence C(oo) =< Card(T(oo)) = > aleph_0. I have no idea, because I have no idea what that cross section is. -- 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 4 Mar 2007 04:01 On 4 Mrz., 02:55, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1172919142.768602.212...(a)z35g2000cwz.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > ... > > > > The cross section of the tree is the same as the number of unit > > > > fractions in the last parenthesis used by Oresme. > > > > > > 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. > > > It is not defined, but all > > parentheses are required to exist. > > It is not defined because the last one does not exist. In any case we can state that there is no parenthesis with more than finitely many elements in it. We can enumerate the elements of every parenthesis by {1} {1,2} {1,2,3,4} .... and we can take the union of all of these sets, being sure that the cardinal number of the union is countable. > > 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. Regards, WM
From: Virgil on 4 Mar 2007 16:38 In article <1172998862.730699.246980(a)n33g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 4 Mrz., 02:55, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1172919142.768602.212...(a)z35g2000cwz.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > ... > > > > > The cross section of the tree is the same as the number of unit > > > > > fractions in the last parenthesis used by Oresme. > > > > > > > > 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. Unless WM allows that the number of terms, either normally or when parenthesized, is infinite, WM doses not know that number. > > > > > It is not defined, but all > > > parentheses are required to exist. > > > > It is not defined because the last one does not exist. > > In any case we can state that there is no parenthesis with more than > finitely many elements in it. > We can enumerate the elements of every parenthesis by > {1} > {1,2} > {1,2,3,4} > ... > and we can take the union of all of these sets, being sure that the > cardinal number of the union is countable. > > > > 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. And when the tree does not have a "last lever". what then little man? > 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. But that procedure does not "count" the set of all paths possible in a complete infinite binary tree, as it only counts paths that are eventually "constant", having either a finite number of left branches or a finite number of right branches. While this is an infinite binary tree, it is no more complete that is the set of all binary rationals (having denominators of form 2^n for some natural n) a complete set of reals. That set of binary rationals is not even a complete set of rationals.
From: Dik T. Winter on 4 Mar 2007 21:08
In article <1172917355.563795.27250(a)n33g2000cwc.googlegroups.com> mueckenh(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: > 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. > > > In order to get to higher ordinals, Hrbacek aqnd Jech say: It is easy > > > to continue the process after this "limit" step is made. > > > > The meaning of 'limit' here is soemthing different, that is why it is in > > quotes. > > > Yes, it is an extension, like the irrational "numbers" are an > extension of the natural numbers. The latter quotation marks are not > necessary, according to you. Wh yare the former? The latter are not necessary, as there is no definition of "numbers". The former are necessary because there is a well known definition of "limit". > > > In fact there are limits in set theory. "Es ist sogar erlaubt, sich > > > die neugeschaffene Zahl omega als Grenze zu denken, welcher die Zahlen > > > nu zustreben, wenn darunter nichts anderes verstanden wird, als da� > > > omega die erste ganze Zahl sein soll, welche auf alle Zahlen nu folgt, > > > d. h. gr��er zu nennen ist als jede der Zahlen nu." > > > > No there are no limits defined in set theory. > > See above. See above. > > What this states it that > > you can look at it as such, but that it is not such, because limits are > > not defined. > > They are not well defined. I agree. They are not defined at all. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |