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Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 10, 2013 4:57 AM
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On 10 Jan., 04:52, Zuhair <zaljo...@gmail.com> wrote:
> On Jan 10, 12:23 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> [Snip confused replies]
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> > 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.
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> > 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.
>
> This is not clear, your terms are not defined clearly. So I cannot
> respond to that.
You cannot understand what the set of all rationals of the unit
interval is? You cannot understand what it means to remove the subset
of all non-terminating decimal representations and to write the
remaining set of all terminating decimal representations in a list
with aleph_0 entries? But you want to make us believe that the
infinite set of all finite initial segments is less infinite than the
set of all digits of an irrational number?
I understand that you suffer from a great psychological shock after
discovering that Cantorism is nonsense.
>
> But anyhow your argument on the whole of it fails and fails badly so.
>
> The reason is that
you cannot think
> we can have even at finite basis more n sized
> tuples of m values than n x m.
But it has been proven formally that aleph_0 * aleph_0 = aleph_0.
Further everything in Cantor's list happens at a finite place. And
finite is less than infinite or aleph_0
So stop your clumsy handwaving and try to recover.
Try again: Consider the list of all terminating decimal
representations.
For every n we have:
The anti-diagonal up to line n must be different from every entry up
to line n.
The anti-diagonal up to digit n must have a double up to digit n.
Result: The diagonal cannot be an entry of the list.
The diagonal cannot be different (at a finite position) from all
entries of the list (and elesewhere, after all finite positions, it
cannot be anything at all).
This is a contradiction.
Regards, WM
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