Another AC anomaly?
in [Logic]
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From: Ross A. Finlayson on 30 Nov 2009 00:29 On Nov 29, 6:54 pm, Marshall <marshall.spi...(a)gmail.com> wrote: > On Nov 29, 5:32 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> > wrote: > > > > > On Nov 28, 8:38 pm, Virgil <Vir...(a)home.esc> wrote: > > > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > On Nov 28, 12:43 pm, Virgil <Vir...(a)home.esc> wrote: > > > > > > Given a specific list of endless binary sequences, the so called Cantor > > > > > diagonal is the result of a specific and unambiguous algorithm applied > > > > > to that list, so it is, for any given list, unique, and not a member of > > > > > the list from which it is constructed. > > > > > > Which WM would have known if he had any sense. > > > > > Binary sequences aren't unique representations of real numbers. > > > > The original Cantor diagonal argument did not deal with real numbers > > > either, so what is your point? > > > The point was that your hasty overgeneralization was false and that it > > represents in your non-acknowledgment hypocritical criticism. > > You haven't identified any mistake of Virgil's. You've merely made > the well-worn point that some real numbers have multiple digit > strings. > > > > > (Binary and ternary (trinary) anti-diagonal cases require refinement.) > > > > But as neither I nor Cantor were not dealing with numbers in any base, > > > your objections are, as usual, irrelevant. > > > No, it was just noted a specific constructive counterexample to that > > lists of (expansions representing) real numbers don't contain their > > antidiagonals. > > It wasn't even that. > > Marshall In binary or ternary an everywhere-non-diagonal isn't not on the list. Using AC, in ZFC, given a well-ordering of the reals, I described a symmetry based construction of a distribution of the natural integers at uniform random. EF is its CDF. Yeah and I am familiar with the other fundamental results of transfinite cardinals and show how they don't hold in nonstandard frameworks suitable to represent the number system for application. Ross F.
From: Virgil on 30 Nov 2009 02:53 In article <3bc52c73-77ee-479b-9a28-824fd9a4a21c(a)z10g2000prh.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > > > > > (Binary and ternary (trinary) anti-diagonal cases require refinement.) > > > > > > But as neither I nor Cantor were not dealing with numbers in any base, > > > > your objections are, as usual, irrelevant. > > > > > No, it was just noted a specific constructive counterexample to that > > > lists of (expansions representing) real numbers don't contain their > > > antidiagonals. > > > > It wasn't even that. > > > > Marshall > > In binary or ternary an everywhere-non-diagonal isn't not on the list. What does "an everywhere-non-diagonal" mean? And does "isn't not on the list" mean the same as "is on the list"? In any integer base, from 2 on up, and any list, there are constructably as many non-members of the list as members of it. > > Using AC, in ZFC, given a well-ordering of the reals Since no one has yet been able to give an explicit well ordering of the reals, we won't give it to you. > I described a > symmetry based construction of a distribution of the natural integers > at uniform random. Is that supposed to mean something in English? [Further garbage DELETED]
From: WM on 30 Nov 2009 06:10 On 29 Nov., 18:35, William Hughes <wpihug...(a)hotmail.com> wrote: > On Nov 29, 8:05 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > WM has conceded that you can use induction > to show that every element of the list has > a final 1, and that there is a constructive > proof that the diagonal number does not have a > final 1. > > WM has a new argument. > > > Use induction to show that the diagonal number cannot have more digits > > than every entry of the list. > > This cannot be done. All you do is show that > every one of an infinite number of different > numbers, none of which is the diagonal number, > cannot have more digits than every entry of the list. There is a simple proof by contradiction: Assume that the diagonal (in the example-list 0.0 0.1 0.11 0.111 ....) has a digit that is not in an entry of the list. This would mean that the list has an end. That is wrong by definition. Therefore every seqeunce of 1's in the diagonal is in an entry of the list. Regards, WM
From: William Hughes on 30 Nov 2009 06:39 On Nov 30, 7:10 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 29 Nov., 18:35, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > On Nov 29, 8:05 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > WM has conceded that you can use induction > > to show that every element of the list has > > a final 1, and that there is a constructive > > proof that the diagonal number does not have a > > final 1. But despite this, in Wolkenmuekenheim, the diagonal number is an element of the list. <snip usual nonsense about the diagonal containing a digit that is not in any entry of the list> - William Hughes
From: Dik T. Winter on 30 Nov 2009 08:39
In article <a934cb81-2667-4152-ac03-ccf696902319(a)c34g2000yqn.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 27 Nov., 17:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > Therefore the vase is never empty - > > > > Similar in the sequence 1/n the elements value is never zero. > > Correct. But in the sequence 1,1,1... the limit is 1 with no doubt. Yes, so what? > > You may verify that in the case you proposed lim X_n does exist and is > > equal to the empty set. > > > > Note: > > 1 = lim |X_n| != |lim X_n| = 0 > > but that should not come as a surprise. > > Let one ball rest in the vase in eternity. The limit will be 1. The limit of the sequence of set of balls is the single ball that stays there forever. And so in that case we have: 1 = lim |X_n| = |lim X_n| = 1 and so what? > Consider the sequence 1, 101, 1, 101, 1, 101, ... There is no limit. Right. > Nevertheless the sequence of minima Minima of what? > 1,1,1,... has limit 1 as before. Yes, so what? > Therefore the result of set theory shows that set theory is not > mathematics. I have no idea how you come to that conclusion. I think you are confusing the limit of a sequence of sets (which is a set) and the limit of the sequence of the cardinalities of sets ( which is a cardinality). In general: the limit of the cardinalities is not necessarily the cardinality of the limit, however much you would like that to be the case. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |