```Date: Feb 11, 2013 12:18 AM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

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 onediagonal for him.  Given n, WM must find a diagonal(note the indefinite article) such that length(dFIS)>n+1so that comparison with the n-th listed sequence canbe made.While there may be other sources for the definitionof "distinguishability", the one I have is in a bookon automata.  Distinguishability is characterized interms of finitary "experiments of length k".  Two"states" are k-distinguishable if there is an experimentof length k which differentiates them.  Two statesare distinguishable if they are k-distinguishablefor any k.Two "states" are k-equivalent if there is no m<=k forwhich the given states are differentiated by an experimentof length m.Two "states" are equivalent if for every k they arenot k-distinguishable.  So, equivalence is infinitary.This description coincides with your explanationas the Cantor diagonal is formed specifically tobe k-distinguishable for every k.As for WM, definite articles imply representationwith singular terms.  He has a plural multiplicityof diagonals.
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