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

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mueckenh@rz.fh-augsburg.de

Posts: 18,076
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Nov 27, 2009 12:36 PM
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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.

> and talking about these ideas generally leads
>> 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


Date Subject Author
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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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