Date: Feb 11, 2013 2:58 AM Author: fom Subject: Re: Matheology § 222 Back to the root<br> s On 2/11/2013 1:53 AM, Virgil wrote:

> 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"?

"some" is certainly better

while writing I must have had "any (particular)"

in mind

>>

>> 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.

>

Exactly.