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 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.