In article <i6adnS7u2oO04oXMnZ2dnUVZ_uWdnZ2d@giganews.com>, fom <fomJUNK@nyms.net> wrote:
> On 2/10/2013 6:30 PM, Virgil wrote: > > In article <hoqdnWmpiaOtvYXMnZ2dnUVZ_s2dnZ2d@giganews.com>, > > fom <fomJUNK@nyms.net> wrote: > > > >> On 2/10/2013 4:16 PM, Virgil wrote: > >>> In article > >>> <3a8b891b-172f-415f-b4f6-34f988abae5d@e10g2000vbv.googlegroups.com>, > >>> WM <mueckenh@rz.fh-augsburg.de> wrote: > >>> > >>>> On 10 Feb., 18:40, William Hughes <wpihug...@gmail.com> wrote: > >>>>> On Feb 10, 10:51 am, WM <mueck...@rz.fh-augsburg.de> wrote: > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>>> On 9 Feb., 17:36, William Hughes <wpihug...@gmail.com> wrote: > >>>>> > >>>>>>>>> the arguments are yours > >>>>>>>>> and the statements are yours- > >>>>> > >>>>>>>> Of course. But the wrong interpretation is yours. > >>>>> > >>>>>>> How does one interpret > >>>>>>> we have shown m does not exist > >>>>>>> (your statement) > >>>>> > >>>>>>> to mean that > >>>>> > >>>>>>> m might still exist > >>>>> > >>>>>>> ? > >>>>> > >>>>>> TND is invalid in the infinite. > >>>>> > >>>>>> Regards, WM > >>>>> > >>>>> In Wolkenmeukenheim, we can have > >>>>> for a potentially infinite set > >>>>> > >>>>> we know that x does not exist > >>>>> we don't know that x does not exist > >>>>> > >>>>> true at the same time. > >>>> > >>>> Is it so hard to conclude from facts without believing in matheology? > >>>> > >>>> The diagonal of the list > >>>> 1 > >>>> 11 > >>>> 111 > >>>> ... > >>>> > >>>> is provably not in a particular line. > >>>> But the diagonal is in the list, since it is defined in the list only. > >>>> Nothing of the diagonal can be proven to surpass the lines and rows of > >>>> the list. > >>> > >>> It is not that the diagonal "surpasses" any particular line, it is > >>> merely that an appropriately defined "diagonal" is different from each > >>> and every particular line, i.e., does not appear as any line among the > >>> lines being listed. > >> > >> Yes. And the scare quotes are nice. > >> > >> The problem with singular terms means that > >> "diagonal" is, in fact, a plurality of acts > >> of definition. > > > > The Cantor antidiagonal rule, for an actually infinite list of actually > > infinite binary sequences is a quite finite rule : > > > > If the two possible values are 'm' and 'w', then the nth term of the > > diagonal is to be not equal to the nth term of the nth listed sequence, > > meaning that > > if the nth term of the nth listed sequence is "m" > > then the nth listed element of the diagonal is "w" > > and > > if the nth term of the nth listed sequence is "w" > > then the nth listed element of the diagonal is "m". > > > > In this way, the constructed sequence differs from the nth listed > > sequences at lest at its nth postion > > > > > Thanks, I do understand that. > > I was referring to WM's position. There cannot be one > diagonal for him. Given n, WM must find a diagonal > (note the indefinite article) such that length(dFIS)>n+1 > so that comparison with the n-th listed sequence can > be made. > > While there may be other sources for the definition > of "distinguishability", the one I have is in a book > on automata. Distinguishability is characterized in > terms of finitary "experiments of length k". Two > "states" are k-distinguishable if there is an experiment > of length k which differentiates them. Two states > are distinguishable if they are k-distinguishable > for any k.
Shouldn't that be "k-distinguishable for some k"? > > Two "states" are k-equivalent if there is no m<=k for > which the given states are differentiated by an experiment > of length m. > > Two "states" are equivalent if for every k they are > not k-distinguishable. So, equivalence is infinitary. > > This description coincides with your explanation > as the Cantor diagonal is formed specifically to > be k-distinguishable for every k. > > As for WM, definite articles imply representation > with singular terms. He has a plural multiplicity > of diagonals. No one of which is the real one. --
