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.