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Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 9, 2013 10:52 PM
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On Jan 10, 12:23 am, WM <mueck...@rz.fh-augsburg.de> wrote:
[Snip confused replies]
>
> Try to read the literature of great mathematicians, for instance
> matheology § 187. Or Bernays: If we pursue the thought that each real
> number is defined by an arithmetical law, the idea of the totality of
> real numbers is no longer indispensable. Or Weyl: Die möglichen
> Kombinationen endlichvieler Buchstaben bilden eine abzählbare Menge,
> und da jede bestimmte reelle Zahl sich durch endlichviele Worte
> definieren lassen muß, kann es nur abzählbar viele reelle Zahlen geben
> - im Widerspruch mit Cantors klassischem Theorem und dessen Beweis. Or
> Schütte (Hilbert's last student): Definiert man die reellen Zahlen in
> einem streng formalen System, in dem nur endliche Herleitungen und
> festgelegte Grundzeichen zugelassen werden, so lassen sich diese
> reellen Zahlen gewiß abzählen, weil ja die Formeln und die
> Herleitungen auf Grund ihrer konstruktiven Erklärungen abzählbar
> sind.
>
> Or try to think with your own head:
> Construct a list of all rationals of the unit interval. Remove all
> periodic representations and keep only the finite ones, but all.
> Then the list contains all finite initial segments which a possible
> anti-diagonal can have. So the anti-diagonal can at no finite position
> deviate from every entry of the list. That means it can never deviate
> from every entry. According to Cantor, anti-diagonal deviates from
> every entry of the list at a finite position. But even according to
> Cantor it deviates never(at no finite position) from all entries.
>
> Contradiction of Cantor.
>
> Regards, WM
This is not clear, your terms are not defined clearly. So I cannot
respond to that.
But anyhow your argument on the whole of it fails and fails badly so.
The reason is that we can have even at finite basis more n sized
tuples of m values than n x m. In the above example of shirts I can
even discriminate 7 persons from wearing THREE kinds of shirts and
only permitting Two wearings only. See this
{(R,R), (Y,Y), (G,G), (R,G), (Y,G), (G,R),(R,Y)}
This intuitively breaks down the distinguishability argument
altogether even at finite level.
If we take the size of a distinguishing tuple to be the
"distinguishability TIME" and the number of possible values of an
entry in each tuple to be the "distinguishability GIRTH", then even at
finite level we can distinguish MORE and MORE tuples than the product
of girth by time of distinguishability. With the case of the reals we
have countably infinite girth and countably infinite time, but yet we
can as demonstrated distinguish MORE distinguishability tuples than
the product of girth by time of distinguishability, i.e. More than
countably many. And since the number of the reals correspond to the
number of distinguishability tuples, then this can be More than
countable. So there is no basis whatsoever at intuitive or formal
level for countability of reals from a distinguishability argument or
the alike reasoning! PERIOD.
Zuhair
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