An uncountable countable set
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From: Franziska Neugebauer on 14 Jul 2006 10:11 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. > 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? > 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. So it is not valid. If you drop it, the contradiction vanishes. 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? 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. non sequitur. BTW: What exactly do you want to show? F. N. -- xyz
From: Dik T. Winter on 14 Jul 2006 10:46 In article <I31cl7cJwktEFwBB(a)212648.invalid> David Hartley <me9(a)privacy.net> writes: > In message <1152785777.506058.199660(a)75g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de writes > > > >David Hartley schrieb: > > > >> The transpositions are relevant in so far as WM claims that Cantor wrote > >> that a sequence of transpositions can not change the order-type. Either > >> Cantor was wrong (or using a different idea of the limit), or WM is > >> misinterpreting him, my German is not good enough to tell. > > > >Daraus folgt, da? solche Umformungen einer wohlgeordneten Menge die > >Anzahl derselben unge?ndert lassen, welche sich auf eine endliche oder > >unendliche Folge von Transpositionen von je zwei Elementen > >zur?ckf?hren lassen, ... > > > >It follows that only such transformations of a well-ordered set leave > >its (ordinal) number unchanged, which can be derived from a finite or > >infinite sequence of transpositions of each two elements, ... > > "of each two elements". That is a strange expression in this context. > ",each of two elements" would make more sense, although redundant unless > he hadn't previously defined "transposition". If that is the case, it > would seem Cantor was wrong, you should have no trouble believing that. It depends also on how transformations are defined. On well-ordered sets there can certainly be transformation that involve infinitely many elements that do not change the ordinal number. Consider a set with order type w*w. Also consider the transformation that interchanges the first two elements of each maximal subsequence of order type w. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 14 Jul 2006 15:26 In article <1152882211.516443.276870(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > > In "usual" mathematics we have 10 * 0.999... = 9.999... and from that > > > we get easily 0.999... = 1. > > > > > > I argue that 0.111... does not exist. > > > > If by "usual mathematics" you mean a mathematics in which the > > meaning of infinite decimals have not been defined, then we > > don't have 0.999... or 9.999... either. > > In usual mathematics 0.999... is assumed to exist. > > > > When you say 0.111... does not exist, do you mean > > > > (a) the notation is undefined, or > > (b) when you define it, you get a number that doesn't exist? > > > You get a representation that does not exist of a number that des exist > as 1/9. If one understands non-terminating decimals to represent the limits of convergent infinite series, as is usual in mathematics, then 0.999... in both senses does exist, both as a representation (more formally as the abbreviation of a representation) of a geometric series and as the limit of that series . > > > 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 > ... > > What is the result? 0.111... As your "addition" schema does not define "infinite" sums, nor the sum of a blank space with a non-blank space, you do not get anything at all when you attempt to "add" all the members of your endless list.
From: Virgil on 14 Jul 2006 15:35 In article <1152882446.397909.108070(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Hartley schrieb: > > > In message <1152785777.506058.199660(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de writes > > > > > >David Hartley schrieb: > > > > > >> The transpositions are relevant in so far as WM claims that Cantor wrote > > >> that a sequence of transpositions can not change the order-type. Either > > >> Cantor was wrong (or using a different idea of the limit), or WM is > > >> misinterpreting him, my German is not good enough to tell. > > > > > >Daraus folgt, da? solche Umformungen einer wohlgeordneten Menge die > > >Anzahl derselben unge?ndert lassen, welche sich auf eine endliche oder > > >unendliche Folge von Transpositionen von je zwei Elementen > > >zur?ckf?hren lassen, ... > > > > > >It follows that only such transformations of a well-ordered set leave > > >its (ordinal) number unchanged, which can be derived from a finite or > > >infinite sequence of transpositions of each two elements, ... > > > > "of each two elements". That is a strange expression in this context. > > That is due to my bad translation. Cantor stresses that each > transposition involves two elements. (Also the "only" above should have > been omitted.) > > > ",each of two elements" would make more sense, although redundant unless > > he hadn't previously defined "transposition". If that is the case, it > > would seem Cantor was wrong, you should have no trouble believing that. > > > Cantor was very often redundant. > > > >> In any case, > > >> it's a proposition WM needs to prove to complete his argument, and he > > >> hasn't done so. > > > > > >What should that be good for? > > > > You would have finally proven a contradiction within standard set > > theory. You've been trying to do that for many years, why give up now, > > when you're so close? All you need to do is prove that, in this > > quotation, Cantor was right! > > Listen to Virgil. If the transpositions would lead to a dense order, > one of them must be the first. But no transposition can be the first. > It is the same as with the Leibniz-series: No fraction finishes it. > Always we have a rational number. There is no limit. pi/4 does not > exist. The distinction here is that for a series of numbers, one has a defined limit process by which one can determine whether a given result is the limit of that infinite process without having to "complete" the process. Since there has been no such "limit process" defined for "mueckenh"'s sequence of transpostions, he cannot do the same. > > > > > > > > >In the binary tree everyone can see that no path can split without two > > >additional edges. So it is by no means possible that there were more > > >paths than edges. Nevertheless the set theorists go through eyes wide > > >shut and do not care. > > > > > >Who would accept logic arguments if set theory was concerned? > > > > Most of us, if you would just provide one. > > No path in the binary tree can split without two more edges - even in > the infinite (which always remains finite, by the way). That should be > sufficient. When one reads such "mueckenh" statements as the oxymoronic "even in the infinite (which always remains finite, by the way)" one easily sees that "mueckenh" is not operating with a full deck.
From: Virgil on 14 Jul 2006 15:40
In article <1152882648.516740.227380(a)35g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > It is done in zero time. > > nonsense. Once the Cantor rule has been stated and vetted, it takes no more time to establish that the number described must differ from any and every number listed. |