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 <>,
> fom <> wrote:

>> On 2/10/2013 4:16 PM, Virgil wrote:
>>> In article
>>> <>,
>>> WM <> wrote:

>>>> On 10 Feb., 18:40, William Hughes <> wrote:
>>>>> On Feb 10, 10:51 am, WM <> wrote:

>>>>>> On 9 Feb., 17:36, William Hughes <> 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.