Virgil
Posts:
4,482
Registered:
1/6/11
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Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 10, 2013 5:04 PM
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In article
<67016130-6977-476b-8333-1df287bccca9@j4g2000yqh.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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]
> >
> >
> >
> >
> >
> >
> >
> > > 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.
> >
> > 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?
What he, and I, find incomprehensible is why
"the anti-diagonal can at no finite position
deviate from every entry of the list"
is presumed to imply
"That means it can never deviate from every entry."
What prevents the diagonal from differing from entry n at position n,
and thus differing from each entry?
That is the way Cantor described it, and nothing WM has said shows why
Cantor was not correct.
> 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.
That "understanding" is typical WMytheology, and outside WMytheology is
false.
>
> But it has been proven formally that aleph_0 * aleph_0 = aleph_0.
But not 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
But there are infinitely many finite places just as there are infinitely
many finite natural numbers.
> So stop your clumsy handwaving and try to recover.
It is WM whose handwaving is clumsy here, as usual!
>
> Try again: Consider the list of all terminating decimal
> representations.
Try to list all NONterminating decimal representations.
You will fail!
Further garbage snipped!
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