Another AC anomaly?
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From: Virgil on 30 Nov 2009 00:21 In article <92659bb1-504c-4c08-8b4e-8aac4fe40619(a)a39g2000pre.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > On Nov 28, 8:38�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <e70292e5-e110-48dc-afda-8ff08c448...(a)g1g2000pra.googlegroups.com>, > > �"Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > > > > > On Nov 28, 12:43�pm, Virgil <Vir...(a)home.esc> wrote: > > > > In article > > > > <f4e15df0-a3c0-48e4-959f-e341a9adf...(a)j4g2000yqe.googlegroups.com>, > > > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > On Nov 27, 5:24�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > > > > On Nov 27, 3:33�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > With only potential, i.e., not finished infinity, i.e., > > > > > > > > > reasonable > > > > > > > > > infinity, �the diagonal number (exchanging 0 by 1) of the > > > > > > > > > following > > > > > > > > > list can be found in the list as an entry: > > > > > > > > > > > 0.0 > > > > > > > > > 0.1 > > > > > > > > > 0.11 > > > > > > > > > 0.111 > > > > > > > > > ... > > > > > > > > > > Only in Wolkenmuekenheim where the argument goes > > > > > > > > > > � � �Every entry in the list has a fixed last 1 > > > > > > > > � � �The diagonal number does not have a fixed last 1 > > > > > > > > > There is not a fixed last entry > > > > > > > > So, �every entry in the list has a fixed last 1. > > > > > > (We don't need a fixed last entry to say this) > > > > > > We still have > > > > > > > > � Every entry in the list has a fixed last 1 > > > > > > � The diagonal number does not have a fixed last 1 > > > > > > > > > Every diagonal number is in the list. > > > > > > > > Only in Wolkenmuekenheim. �Outside of Wolkenmuekenheim > > > > > > there is only one diagonal number > > > > > > > How do you know, unless you have seen the last? > > > > > > > Regards, WM > > > > > > 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. > > > > (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. > > > > > > > > For example, the list contains .1 then all zeros, the anti-diagonal > > > is .0111... = .100..., anti-diagonal is on the list. > > > > But, in the Cantor argument, the lists in question are not of functions > > from N to range {0,1} but of functions from N to range {m,w} with no > > assumption that such a function corresponds to any sort of number. > > > > It was simply an example that a list of real numbers contains its anti- > diagonal because of dual representation of some standard Eudoxus/ > Dedekind/Cauchy fixed radix expansion expressions of a real number. Even when one assumes the Cantor diagonalization is to be applied to ists of binary numerals instead lists of of arbitrary binary sequences, there are well known ways of avoiding the dual representation problem. Which Ross has seem often enough s that he should be familiar with them, unless his memory is faulty. > > > So Ross is, as usual, in over his head. > > > > > > No. Why are you trying to bait and switch instead of simply > acknowledging that particular lambast was mistaken? As I am referring to the original Cantor argument rather than any of the later modifications of it as Ross is doing, it is Ross who is doing a bait and switch.
From: Ross A. Finlayson on 30 Nov 2009 00:29 On Nov 29, 9:21 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <92659bb1-504c-4c08-8b4e-8aac4fe40...(a)a39g2000pre.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > On Nov 28, 8:38 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <e70292e5-e110-48dc-afda-8ff08c448...(a)g1g2000pra.googlegroups.com>, > > > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > On Nov 28, 12:43 pm, Virgil <Vir...(a)home.esc> wrote: > > > > > In article > > > > > <f4e15df0-a3c0-48e4-959f-e341a9adf...(a)j4g2000yqe.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > On Nov 27, 5:24 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > > > > > On Nov 27, 3:33 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > > > > > > > With only potential, i.e., not finished infinity, i.e., > > > > > > > > > > reasonable > > > > > > > > > > infinity, the diagonal number (exchanging 0 by 1) of the > > > > > > > > > > following > > > > > > > > > > list can be found in the list as an entry: > > > > > > > > > > > 0.0 > > > > > > > > > > 0.1 > > > > > > > > > > 0.11 > > > > > > > > > > 0.111 > > > > > > > > > > ... > > > > > > > > > > Only in Wolkenmuekenheim where the argument goes > > > > > > > > > > Every entry in the list has a fixed last 1 > > > > > > > > > The diagonal number does not have a fixed last 1 > > > > > > > > > There is not a fixed last entry > > > > > > > > So, every entry in the list has a fixed last 1. > > > > > > > (We don't need a fixed last entry to say this) > > > > > > > We still have > > > > > > > > Every entry in the list has a fixed last 1 > > > > > > > The diagonal number does not have a fixed last 1 > > > > > > > > > Every diagonal number is in the list. > > > > > > > > Only in Wolkenmuekenheim. Outside of Wolkenmuekenheim > > > > > > > there is only one diagonal number > > > > > > > How do you know, unless you have seen the last? > > > > > > > Regards, WM > > > > > > 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. > > > > > (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. > > > > > For example, the list contains .1 then all zeros, the anti-diagonal > > > > is .0111... = .100..., anti-diagonal is on the list. > > > > But, in the Cantor argument, the lists in question are not of functions > > > from N to range {0,1} but of functions from N to range {m,w} with no > > > assumption that such a function corresponds to any sort of number. > > > It was simply an example that a list of real numbers contains its anti- > > diagonal because of dual representation of some standard Eudoxus/ > > Dedekind/Cauchy fixed radix expansion expressions of a real number. > > Even when one assumes the Cantor diagonalization is to be applied to > ists of binary numerals instead lists of of arbitrary binary sequences, > there are well known ways of avoiding the dual representation problem. > > Which Ross has seem often enough s that he should be familiar with them, > unless his memory is faulty. > > > > > > So Ross is, as usual, in over his head. > > > No. Why are you trying to bait and switch instead of simply > > acknowledging that particular lambast was mistaken? > > As I am referring to the original Cantor argument rather than any of the > later modifications of it as Ross is doing, it is Ross who is doing a > bait and switch. Quote: > > > 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.
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 |