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Topic: Matheology � 233
Replies: 36   Last Post: Apr 2, 2013 5:56 PM

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fom

Posts: 1,969
Registered: 12/4/12
Re: Matheology § 233
Posted: Mar 30, 2013 9:55 PM
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On 3/30/2013 7:56 PM, Virgil wrote:
> In article
> <2bc13fff-5cbb-43dd-a06a-218c68c99849@m9g2000vbc.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> On 30 Mrz., 22:11, Virgil <vir...@ligriv.com> wrote:
>>

>>>>> Thus there is always at least one bit of any listed entry disagreeing
>>>>> with the antidiagonanl, just as the Cantor proof requires.

>>>
>>>> In a list containing every rational: Is there always, i.e., up to
>>>> every digit, an infinite set of paths identical with the anti-
>>>> diagonal? Yes or no?

>>>
>>> The set of paths in any Complete Infinite Binary Tree which agree with
>>> any particular path up to its nth node is equinumerous with the set of
>>> all paths in the entire tree i.e., is uncountably infinite.

>>
>> This was the question: In a list containing every rational: Is there
>> always, i.e., up to every digit, an infinite set of paths identical
>> with the anti-diagonal? Yes or no?

>
> Lists and trees are different. And anti-diagonals derive from lists, not
> trees.
> The entries in list are well ordered.
> The entries in a Complete Infinite Binary Tree are densely ordered.
> Those order types are incompatible.
> So questions, like WM's, which confuse them, are nonsense.
> At least outside Wolkenmuekenheim.
>


But note that the question also demonstrates WM's
complete lack of understanding of the diagonal
argument.

He has been told time and time again that it is
an argument scheme which only has application
under certain assumptions.

He chooses to believe otherwise for the agenda
of his fanaticism.

Suppose one is given a countable listing of
the rationals (with the appropriate restriction
on double representation) according to the
infinite listing of an expansion.

Suppose one performs a diagonalization on
that listing.

Is the resultant a rational number? No.

What may be concluded? That the rational numbers
do not exhaust the capacity of the algorithm
to generate representations if that algorithm
is to generate a representation for every
rational number.

Although the burden of proof lies with WM
concerning the nature of the diagonal, it
is a simple matter to understand if one
uses a Baire space representation instead.

In the Baire space, rationals are in correspondence
with eventually constant sequences. Since,
by construction, a list of Baire space rationals
would exhaust all of the eventually constant
sequences, the resultant of a diagonal argument
could not have an eventually constant sequence
unless the original premise had been false.

















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