Another AC anomaly?
in [Logic]
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From: WM on 28 Nov 2009 07:35 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
From: William Hughes on 28 Nov 2009 08:11 On Nov 28, 8:35 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 27 Nov., 22:42, William Hughes <wpihug...(a)hotmail.com> wrote: > > > Every diagonal number is in the list. > > > Only in Wolkenmuekenheim. Outside of Wolkenmuekenheim > > there is only one diagonal number and it is not > > in the list. > > How do you know, unless you have seen the last? > You use induction to show that every entry in the list has a final 1. So you don't have to see an entry to know that it has a final 1 (there is no last entry in any case). There is a constructive proof that the diagonal number does not have a final 1. - William Hughes
From: Virgil on 28 Nov 2009 15:43 In article <f4e15df0-a3c0-48e4-959f-e341a9adf3ef(a)j4g2000yqe.googlegroups.com>, WM <mueckenh(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.
From: Ross A. Finlayson on 28 Nov 2009 22:24 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. (Binary and ternary (trinary) anti-diagonal cases require refinement.) For example, the list contains .1 then all zeros, the anti-diagonal is .0111... = .100..., anti-diagonal is on the list. Ross F.
From: Virgil on 28 Nov 2009 23:38
In article <e70292e5-e110-48dc-afda-8ff08c448dbd(a)g1g2000pra.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(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? > (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. > > 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. So Ross is, as usual, in over his head. > > Ross F. |