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Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF
CANTOR: THE REALS ARE UNCOUNTABLE!
Posted:
Jan 10, 2013 2:21 AM
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On Jan 9, 7:52 pm, 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.
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> > Contradiction of Cantor.
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> > Regards, WM
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> This is not clear, your terms are not defined clearly. So I cannot
> respond to that.
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> But anyhow your argument on the whole of it fails and fails badly so.
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> 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
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> {(R,R), (Y,Y), (G,G), (R,G), (Y,G), (G,R),(R,Y)}
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> This intuitively breaks down the distinguishability argument
> altogether even at finite level.
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> 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
Consider then the difference between the funnel and the sieve. One
might pour the media into the funnel and differentiate each as the
fall out, and the funnel would fill, a simple rate problem. The sieve
then might simply differentiate across at once, via the structure
being itself defined in the parallel: instead of bringing each item
to the distinguishment, a sweep through them differentiates them at
once.
Al Jofar, I appreciate that you find profundity in the uncountability
of the regular, well-founded powerset of the regular, well-founded
infinite set, yet as well you know of development with infinite
elements in the infinite set of finite things, or of the anti-
foundational, or expressly of how today's theorists grapple with the
most fundamentally large, the universe or the totality of ordinals,
which are differentiated via their structure more strongly than
extensionality. You know that extensionality and for that matter the
particular axiomatization of infinity as regular and well-founded,
restricted, is just that: an assertion of an abstract thing. Courtesy
the lack of universal quantification over the inductively following
elements, it is known that theory is either incomplete, or
inconsistent. At best consistent, there are plain true facts about
its objects that aren't of its theorems, but the structure of each is
declared.
Russell, for example, noted that the set of all sets that don't
contain themselves, would contain itself. That's simple: it does.
And, that's ZF's universe.
So, while on the one hand it makes sense that you state your belief in
the validity of the uncountable, of said theory's axioms and its
immediately accessible theorems courtesy extensionality and a
particularly declared restrictedly regular infinity, because you
describe theories generally and feel it reasonable to express to your
audience your belief directly, as well there are simply theories
without those axioms and you well know that.
Then, with regards to attempts to understand and rigorously define
mathematical abstractions of the continuum, of numbers, and geometry,
and simply collections or grouping and separation, obviously enough
everyone goes about that as part of reason. Then with regards to that
"only finites make an infinite" but then "infinites don't make an
infinite", it's not necessarily ill-conceived to find alternate
axioms, or particularly a lack thereof, see different true results,
then to the understanding and distinction of mathematical
abstractions.
Then, where I'm particularly aggressive and instead of simply defining
structures in vacuo, because they're not, instead carry forward the
general case of working toward their conciliation, of the universal
and empty, I go about that constructively, for mathematics for
humanity and not just mathematicians for themselves: for the applied.
The universe as set is its own powerset. I find it easier to believe
that the all exists, than that it doesn't.
Because, otherwise, it wouldn't.
Regards,
Ross Finlayson
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