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.
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.
