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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Another AC anomaly?
Posted: Nov 27, 2009 2:33 PM

On 27 Nov., 18:55, A <anonymous.rubbert...@yahoo.com> wrote:
> On Nov 27, 12:36 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
>
>
>
>

> > On 27 Nov., 16:21, A <anonymous.rubbert...@yahoo.com> wrote:
>
> > > On Nov 27, 1:43 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > On 27 Nov., 03:50, A <anonymous.rubbert...@yahoo.com> wrote:
>
> > > > > On Nov 26, 3:51 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > On 26 Nov., 19:22, William Hughes <wpihug...@hotmail.com> wrote:
>
> > > > > > > On Nov 26, 12:24 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> > > > > > > > Here is another interesting task: Use balls representing the positive
> > > > > > > > rationals. The first time fill in one ball. Then fill in always 100
> > > > > > > > balls and remove 100 balls, leaving inside the ball representing the
> > > > > > > > smallest of the 101 rationals.

>
> > > > > > > [at random with any measure that gives a positive probability
> > > > > > > to each rational]

>
> > > > > > Simply take the first, seconde, third ... Centuria according to
> > > > > > Cantor's well-ordering of the positive rationals. Then there is no
> > > > > > need for considering any probabilities.

>
> > > > > > > > If you get practical experience, you
> > > > > > > > will accomplish every Centuria in half time. So after a short while
> > > > > > > > you will have found the smallest positive rational.

>
> > > > > > > Only in Wolkenmuekenheim.  Outside of Wolkenmuekenheim
> > > > > > > you will have an empty set.

>
> > > > > > Besides your assertion, you have arguments too, don't you?
> > > > > > In particular you can explain, how the empty set will emerge while
> > > > > > throughout the whole time the minimum contents of the vase is 1 ball?

>
> > > > > > Regards, WM
>
> > > > > Let S denote a set with exactly 101 elements. Let Q+ denote the
> > > > > positive rational numbers. Let inj(S,Q+) denote the set of injective
> > > > > functions from S to Q+. Let {x_n} denote a sequence of elements of inj
> > > > > (S,Q+) with the following properties:

>
> > > > > 1. Let im x_n denote the image of x_n. Then the union of im x_n for
> > > > > all n is all of Q+.

>
> > > > > 2. For any n, the intersection of im x_n with im x_(n+1) consists of
> > > > > exactly one element, which is the minimal element (in the standard
> > > > > ordering on Q+) in im x_n.

>
> > > > > Let X denote the subset of Q+ defined as follows: a positive rational
> > > > > number x is in X if and only if there exists some positive integer N
> > > > > such that, for all M > N, x is in the image of x_M.

>
> > > > > We are talking about X, right?
>
> > > > We are talking about a vase which is never emptied completely!
>
> > > > Hence it cannot be empty unless "infinity" is identical to "never".
> > > > But this describes potential infinity and excludes phantasies like
> > > > Cantor's finished diagonal number.

>
> > > > Regards, WM
>
> > > > Regards, WM
>
> > > The set X described above is certainly the empty set, as one can
> > > easily prove, despite im x_n being nonempty for each n. This seems to
> > > be a rigorous statement of what you are describing by talking about
> > > balls, vases, etc. I have never seen any mention of "potential
> > > infinity" or "completed infinity" which is precise enough to even be
> > > considered mathematics,

>
> > Completed infinity is the same as actual infinity. You need it to
> > obtain a diagonal number. Without that the number would never get
> > finished.

>
> > Potential infinity is the infinity of mathematics. It has always been
> > used until matheology started.

>
> I do not know what "matheology" is

modern logic and set theory, a mixture between mathematics and
theology. The latter is prevailing.

, nor what "completed infinity,"
> "actual infinity," and "potential infinity" are. Do these things
> actually have rigorous definitions?
>

That depends on your understanding of rigor.
>
> > >> to nothing better than confusion--it might be better to speak
> > > precisely about the limits or sets that you mean, in any given
> > > situation, rather than worrying whether they represent "potential
> > > infinity" or "completed infinity

>
> > You can talk about limits but you cannot talk about Cantor's diagonal
> > without actual infinity. If doing so, you get numbers that cannot be
> > named. This is the best proof showing that actual infinity is not to
> > be considered mathematics.

>
> > Regards, WM
>
> I do not know what "Cantor's diagonal" you are speaking of, but the
> method of proving that the real numbers are uncountable which is
> usually called "Cantor's diagonal argument" does not ever mention any
> "actual infinity."

Nevertheless it requires actual infinity.
>
> Perhaps you have some constructivist objections to mathematics as it
> is practiced using ZFC set theory and the nonconstructive proofs it
> allows. If that's so, then fine; Cantor's diagonal argument works fine
> in ZFC but a strict constructivist, who insists on a more restrictive
> set theory or a more restrictive logic, can find reasons to object to
> it.-

A strict constructivist denies actaul infinity and, hence, uncountable
sets.

With only potential, i.e., not finished infinity, i.e., reasonable
infinity, the diagonal number (exchanging 0 by 1) of the following
list can be found in the list as an entry:

0.0
0.1
0.11
0.111
...

Thus Cantor's method fails - here and everywhere.

Regards, WM

Date Subject Author
11/23/09 Jesse F. Hughes
11/23/09 Herman Jurjus
11/23/09 master1729
11/25/09 T.H. Ray
11/24/09 george
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11/30/09 Herman Jurjus
12/1/09 plutonium.archimedes@gmail.com
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11/27/09 William Hughes
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11/28/09 mueckenh@rz.fh-augsburg.de
11/28/09 Virgil
11/26/09 William Hughes
11/28/09 William Hughes
11/26/09 mueckenh@rz.fh-augsburg.de
11/28/09 ross.finlayson@gmail.com
11/28/09 Virgil
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11/29/09 Marshall
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11/30/09 Virgil
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11/30/09 Virgil
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11/30/09 Virgil
11/29/09 mueckenh@rz.fh-augsburg.de
11/26/09 LauLuna
11/26/09 William Hughes
11/26/09 anonymous.rubbertube@yahoo.com
11/29/09 William Hughes
11/27/09 mueckenh@rz.fh-augsburg.de
11/27/09 mueckenh@rz.fh-augsburg.de
11/27/09 Virgil
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11/29/09 William Hughes
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11/27/09 Dik T. Winter
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11/30/09 Dik T. Winter
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12/1/09 Virgil
12/1/09 ross.finlayson@gmail.com
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