From: William Hughes on
On Nov 27, 3:33 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:

> 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
> ...

Only in Wolkenmuekenheim where the argument goes

Every entry in the list has a fixed last 1
The diagonal number does not have a fixed last 1
The diagonal number is an entry in the list


- William Hughes

From: WM on
On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote:
> On Nov 27, 3:33 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
>
> > 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
> > ...
>
> Only in Wolkenmuekenheim where the argument goes
>
>      Every entry in the list has a fixed last 1
>      The diagonal number does not have a fixed last 1
There is not a fixed last entry.
>      [...]
Every diagonal number is in the list.

Regards, WM
From: William Hughes on
On Nov 27, 5:24 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote:
>
> > On Nov 27, 3:33 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
>
> > > 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
> > > ...
>
> > Only in Wolkenmuekenheim where the argument goes
>
> >      Every entry in the list has a fixed last 1
> >      The diagonal number does not have a fixed last 1
>
> There is not a fixed last entry

So, every entry in the list has a fixed last 1.
(We don't need a fixed last entry to say this)
We still have

Every entry in the list has a fixed last 1
The diagonal number does not have a fixed last 1


> Every diagonal number is in the list.

Only in Wolkenmuekenheim. Outside of Wolkenmuekenheim
there is only one diagonal number and it is not
in the list.


- William Hughes

From: Virgil on
In article
<e998c756-bb05-4370-be93-5f811aee443a(a)a21g2000yqc.googlegroups.com>,
WM <mueckenh(a)rz.fh-augsburg.de> wrote:

> On 27 Nov., 18:55, A <anonymous.rubbert...(a)yahoo.com> wrote:
> > On Nov 27, 12:36�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> >
> >
> >
> >
> >
> > > On 27 Nov., 16:21, A <anonymous.rubbert...(a)yahoo.com> wrote:
> >
> > > > On Nov 27, 1:43�am, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> >
> > > > > On 27 Nov., 03:50, A <anonymous.rubbert...(a)yahoo.com> wrote:
> >
> > > > > > On Nov 26, 3:51�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> >
> > > > > > > On 26 Nov., 19:22, William Hughes <wpihug...(a)hotmail.com> wrote:
> >
> > > > > > > > On Nov 26, 12:24�pm, WM <mueck...(a)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.

It is a area which does not exist outside of Wolkenmeukenheim, and does
not include much of either logic or of set theory
>
> , nor what Wolkenmeukenheim are. Do these things
> > actually have rigorous definitions?
> >
> That depends on your understanding of rigor.

Those who understand and apply rigor have no need for any of
Muekenheim's perverted notions of "completed infinity," "actual
infinity," and "potential infinity"
> >
> > > > 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
> >
> > 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.

It only requires a very simple an understanding of "infinite" which is
still way beyond Muekenheim's capabilities.

A non-empty set is finite if for any total ordering of its elements
there is a last element, and is infinite otherwise, i.e., is totally
orderable ssso that it does NOT have a last element.
> >
> > 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.

But mathematics is not in thrall to strict constructionalists.

>
> Thus Cantor's method fails - here and everywhere.

Only within the highly constraining boundaries of Wolkenmuekenheim.

Those who are not so constrained, like Cantor, do not all fail.
From: Virgil on
In article
<ddb474bb-2113-4cff-a104-99f6ceed39ad(a)d21g2000yqn.googlegroups.com>,
WM <mueckenh(a)rz.fh-augsburg.de> wrote:

> On 27 Nov., 21:17, William Hughes <wpihug...(a)hotmail.com> wrote:
> > On Nov 27, 3:33�pm, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> >
> > > 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
> > > ...
> >
> > Only in Wolkenmuekenheim where the argument goes
> >
> > � � �Every entry in the list has a fixed last 1
> > � � �The diagonal number does not have a fixed last 1
> There is not a fixed last entry.
> > � � �[...]
> Every diagonal number is in the list.
>
> Regards, WM

Given any endless list of endless sequences from a two element set, such
as Cantor's {m,w}, at which position in that list is the "anti-diagonal'
to that list, Muekenheim?

Unless Muekenheim has a specific and correct answer, he gets a failing
grade.