An uncountable countable set
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From: mueckenh on 15 Jul 2006 11:56 Dik T. Winter schrieb: > In article <1152884145.033413.104250(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > These words are uninteresting. Transpositions or exchanges of two > > > > elements are clearly defined. Infinitely many are admitted by Cantor. > > > > > > This is wrong. Not any sequence of transpositions is permitted. What > > > *is* permitted is *transformations that can be written as a sequence of > > > transpositions*. Because if you admit *any* sequence of transpositions > > > you may destroy the order-type. > > > > > Works, p. 214: Die Frage, durch welche Umformungen einer wohlgeordneten > > Menge ihre Anzahl geändert wird, durch welche nicht, läßt sich > > einfach so beantworten, daß diejenigen und nur diejenigen Umformungen > > die Anzahl ungeändert lassen, welche sich zurückführen lassen auf > > eine endliche oder unendliche Menge von Transpositionen, d. h. von > > Vertauschungen je zweier Elemente. > > > > No word about sequence or permutation at this place. > > No, but still, not *any* set of transpositions is permitted. Only those > sets of transpositions that are the result of rewriting a transformation > ("Umformungen"). So it remains interesting what Cantor is meaning when > he writes "Umformungen". And when he is meaning with that word arbitrary > re-orderings, he is wrong (because *all* re-orderings can be rewritten as > an infinite set of transpositions). You do completely misunderstand what is defined and what is the defintion. Defined is Umformungen: "diejenigen und nur diejenigen Umformungen = those and only those transformations" they are defined by: "welche sich zurückführen lassen auf eine endliche oder unendliche Menge von Transpositionen, d. h. von Vertauschungen je zweier Elemente = which can be traced back to a finite or infinite set of transpositions, i.e., interchanges of two elements." > > > > > > The reordering of the naturals (0, 1, 2, ...) to (1, 2, 3, ..., 0) > > > > > can be written as an infinite sequence of transpositions, but it > > > > > changes the ordinal number. But is it a permutation? > > > > > > > > Unimportant for Cantor's statement. > > > > > > It *is* important. Because if that falls under Cantor's definition of > > > "transformation" and "permutation", Cantor's statement is trivially false. > > > > Of course it is false, whether there is a sequence or not. My > > transpositions form a sequence. One cannot do better. > > Yes, so what? Is yuor set of transpositions a rewriting of some "Umformung" > in the sense Cantor implies? So there is no "of course" here. We need to > know what Cantor means when he writes "Umformungen". > He says it clearly: those and only those transformations which are realized by a finite or infinite set of transpositions, i.e., interchanges of two elements, do not change the Anzahl = ordinal number. > > > Eh? You assume that Cantor's conclusion is true for your sequence of > > > transpositions. I state that Cantor's conclusion is false for your > > > sequence of statements. > > > > Cantor's conclusion is correct, with no doubt, if infinite sets like > > the set of my transpositions do exist. > > No, when you assume that "Umformungen" means *any* re-ordering, it is > trivially false if infinite sets do exist. But pray *prove* Cantor's > conclusion based on > (1) infinite sets do exist > (2) an "Umformung" is any re-ordering > You state there is no doubt, so it should be easy for you to provide a > proof. He says it: those and only those transformations which are realized by a finite or infinite set of transpositions, i.e., interchanges of two elements. > > > > > I know your interpretation is wrong. > > > > > > What part of my interpretation is wrong? > > > > The implication that Cantor's statement could be correct (if special > > constraints were obeyed by permutations) with the implication that > > infinity could exist. > > Can you provide a proof of that statement? Because you state just above: > "Cantor's conclusion is correct, with no doubt, if infinite sets like > the set of my transpositions do exist." Which contradicts what you state > here. No. Cantor obviously was correct, if infinite sets would exist. But they do not. So he was wrong. And you are wrong, as I have shown above by Cantor's text which you misinterpreted grossly. > > > > Therefore it never becomes 0 but always remains > 0 like 0.11... never > > becomes 1/9. > > But apparently you do not use the standard definition of 0.111... . Is it > a number or something else? It is a number with aleph_0 places, which are all occupied by 1's. in the list: 0.1 0.11 0.111 .... there cannot all places be occupied by 1's. Hence the sum cannot be 0.111... . Regards, WM
From: mueckenh on 15 Jul 2006 12:15 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > [...] > >> If (a), then that is fixed by defining it, which can be done with > >> limits. > > > > It cannot. You see it by the following argument. > > Define 0 *+ 0 = 0, 0 *+ 1 = 1 *+ 0 = 1 *+ 1 = 1. > > Do the following addition *+ > > > > 0.1 > > 0.11 > > 0.111 > > ... > > You /may/ define *+ also as an operator on sequences (like U). > > / > | 0 if A n (n e N & a_n = 0) > *+[a_i] := a_0 *+ a_1 *+ ... := | > | 1 if E n (n e N & a_n = 1) > \ > i e N, a_i e {0, 1} > > You may now assume 0.1 := 0.10... (allocational interpretation) then > you can define the matrix of figures of your "list": > > c_ij is the figure in line i column j. > > To sum up a column j now means *+(summed over i)[c_ij]. > > The result is "collected" by d_j := *+(summed over i)[c_ij]. > > > What is the result? 0.111... > > Yes. d_j = 1 if E n (n e N & c_ij = 1) A j e N. Since c_jj = 1 A j e N > this is true. Every d_i = 1 is on a finite position. Hence every number ends with a zero. The sum cannot be larger. 0.111... is a number with aleph_0 places, which are all occupied by 1's. In the list: 0.1 0.11 0.111 .... there is no number with all places occupied by 1's. Hence the sum cannot be 0.111... . > > > This number is not among the numbers to be added. > > Obviously not. The numbers to be *-added are all the representations > of natural numbers, not the representations of unnatural numbers > (0,111...). > > > But it must be, > > From which presumption do you conclude that? >From the fact that you must pretend it. > > > if it is different from any list number and if the result of the > > addition is 0.111... . > > Your claim is contradictory. If it "must" be in the "list" > then it necessarily "must" be *equal* to some list number > (that is the definition of "in the list"). > > Furthermore your claim is not derived from common presumptions. Summing up zeros gives zero. All numbers of he list end with a zero (even with infinitely many zeros). > So it is not valid. If you drop it, the contradiction vanishes. An addition cannot supply 1's at all places if all summands end with a 10, i.e., one,zero combination. > > You may define the columnwise addtion in a different fashion. > My definition is merely a suggestion. And you are free to disclose > your hidden presumptions you derive your claim from. > > > Its absence does not matter as its presence does not matter. > > You may take any number off the list. Its absence then does not > "matter" (i.e. change the result). So what? If only numbers ending with zero are present, then the calculation must reflect this fact. Or do you sum zeros to get 1? > > You have "removed" the add carry from the ordinary addition to > establish the *-addition (formerly knwon as or-operation). What > are you complaining about? > > > Add by *+ > > > > 0,1 > > 0.11 > > 0.111 > > ... > > 0.111... > > > > The result is the same. Hence, if 0.111... exists, > > it is in he list and is not in the list. > You may add by *+ the list below with the diagonal number exchanged by (0 --> 1). You will see that the diagonal number, whether written down explicitly as in the list above or to be introduced in the list below there is no change of the sum. Hence the diagonal is already in the list. 0.0 0.1 0.11 0.111 .... Regards, WM
From: Franziska Neugebauer on 15 Jul 2006 13:00 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Randy Poe schrieb: >> > [...] >> >> If (a), then that is fixed by defining it, which can be done with >> >> limits. >> > >> > It cannot. You see it by the following argument. >> > Define 0 *+ 0 = 0, 0 *+ 1 = 1 *+ 0 = 1 *+ 1 = 1. >> > Do the following addition *+ >> > >> > 0.1 >> > 0.11 >> > 0.111 >> > ... >> >> You /may/ define *+ also as an operator on sequences (like U). >> >> / >> | 0 if A n (n e N & a_n = 0) >> *+[a_i] := a_0 *+ a_1 *+ ... := | >> | 1 if E n (n e N & a_n = 1) >> \ >> i e N, a_i e {0, 1} >> >> You may now assume 0.1 := 0.10... (allocational interpretation) then >> you can define the matrix of figures of your "list": >> >> c_ij is the figure in line i column j. >> >> To sum up a column j now means *+(summed over i)[c_ij]. >> >> The result is "collected" by d_j := *+(summed over i)[c_ij]. >> >> > What is the result? 0.111... >> >> Yes. d_j = 1 if E n (n e N & c_ij = 1) A j e N. Since c_jj = 1 A j e >> N this is true. > > Every d_i = 1 is on a finite position. There only natural positions (i.e. with positional index n e N). BTW: d_i denotes the Summ of the "list". d does not refer to some "line" of the list. > Hence every number ends with a zero. Do you know what "hence" means? > The sum cannot be larger. 0.111... is a number with aleph_0 > places, which are all occupied by 1's. Every representation of the list is by definition indexed with naturals. > In the list: > > 0.1 > 0.11 > 0.111 > ... > there is no number with all places occupied by 1's. Hence the sum > cannot be 0.111... . still non sequitur. >> > This number is not among the numbers to be added. >> >> Obviously not. The numbers to be *-added are all the representations >> of natural numbers, not the representations of unnatural numbers >> (0,111...). >> >> > But it must be, >> >> From which presumption do you conclude that? > Still can't see your presumptions. >>From the fact that you must pretend it. >> >> > if it is different from any list number and if the result of the >> > addition is 0.111... . >> >> Your claim is contradictory. If it "must" be in the "list" >> then it necessarily "must" be *equal* to some list number >> (that is the definition of "in the list"). >> >> Furthermore your claim is not derived from common presumptions. > > Summing up zeros gives zero. All numbers of he list end with a zero > (even with infinitely many zeros). Every column contains at least one 1. After the definition of the columnwise infinite *-summation (yours and mine) every column thus is *-summed up yielding 1. >> So it is not valid. If you drop it, the contradiction vanishes. > > An addition cannot supply 1's at all places if all summands end with a > 10, i.e., one,zero combination. There is no column not containing a 1. >> You may define the columnwise addtion in a different fashion. >> My definition is merely a suggestion. And you are free to disclose >> your hidden presumptions you derive your claim from. >> >> > Its absence does not matter as its presence does not matter. >> >> You may take any number off the list. Its absence then does not >> "matter" (i.e. change the result). So what? > > If only numbers ending with zero are present, then the calculation > must reflect this fact. Or do you sum zeros to get 1? Once again: There is no column without a 1. /Hence/ every column *-summs up to 1. >> You have "removed" the add carry from the ordinary addition to >> establish the *-addition (formerly knwon as or-operation). What >> are you complaining about? >> >> > Add by *+ >> > >> > 0,1 >> > 0.11 >> > 0.111 >> > ... >> > 0.111... >> > >> > The result is the same. Hence, if 0.111... exists, >> > it is in he list and is not in the list. >> > You may add by *+ the list below with the diagonal number exchanged by > (0 --> 1). You will see that the diagonal number, whether written down > explicitly as in the list above or to be introduced in the list below > there is no change of the sum. Hence the diagonal is already in the > list. > > > 0.0 > 0.1 > 0.11 > 0.111 > ... Are you now going to answer yourself? Please define "is in the list". Uh. Wait. Please define "define" first. F. N. -- xyz
From: Virgil on 15 Jul 2006 15:06 In article <1152978585.858028.263870(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > In usual mathematics 0.999... is assumed to exist. > > > > It is not assumed. There is a definition floating around giving a meaning > > to that notation. > > Without the definition that notation makes no sense. > > And with that definition it makes no sense either, as the definition is > of he same kind as the squared circle. > > > You are confusing notation with the basic concept. > > The notation is the expression of the concept but it is impossible to > separate both. Without an expression there is no concept. Both are > assumed to exist, erroneously. Does "mueckenh" claim that neither the concept nor the expression of a convergent geometric series can exist? In mathematical philosophies such series are as certain to exist as 1,2 and 3, which are themselves, after all, only concepts or expressions. Thus it would appear that whatever philosophy "mueckenh' has, it is anti-mathematical.
From: Virgil on 15 Jul 2006 15:09
In article <1152979000.961294.290560(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > No. Cantor obviously was correct, if infinite sets would exist. But > they do not. If the issue is to be between whether to accept Cantor or to accept "mueckenh", Cantor will prevail. |