Date: Feb 11, 2013 2:53 AM Author: Virgil Subject: Re: Matheology � 222 Back to the roots 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.

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