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Topic: Matheology § 246
Replies: 246   Last Post: Apr 26, 2013 2:37 AM

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mueckenh@rz.fh-augsburg.de

Posts: 14,648
Registered: 1/29/05
Re: Matheology § 246
Posted: Apr 13, 2013 3:29 AM
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On 12 Apr., 23:29, Dan <dan.ms.ch...@gmail.com> wrote:
> > No, but it contains all decimal fractions, i.e., all terminating
> > decimals. There is no chance to represent 1/3 or sqrt(2) by digits.
> > There is no chance to apply Cantor's argument other than to FISs.
> > Every d_n is part of a FIS.

>
> You can represent by an infinite sequence of digits, at least in an
> abstract sense. That's why we can talk about pi, and the questions
> like "what is the 1000000'th digit in the decimal expansion of pi?"
> have a valid unambiguous answer .


Yes, we hav a formula that tells us the n-th digit of pi. But the
*decimal representation of pi* does never leave the domain of FISs. A
decimal string would not allow us to recognize pi. Therefore there is
no decimal representation of pi and of other actually infinite
sequences. Therefore the anti-diagonal d, when only presenting its
digits d_n, cannot differ from all FISs and therefore cannot prove the
uncountability of irrational numbers.

Do you agree that
{1} U {1, 2} U {1, 2, 3} U ... = |N
but the infinite sequence
{1}, {1, 2}, {1, 2, 3}, ...
does not contain |N?

What is the difference?

>
> > Since the Binary Tree deletes the difference between sequence and
> > unions, it shows this problem in full clarity.

>
> > The Binary Tree is constructed by countably many nodes. So it is
> > impossible to distinguish uncountably many paths *by nodes*.

>
> Yes . The binary tree has countable nodes , and uncountable paths .
> (paths sequences of nodes, ie. subsets of the set of nodes , therefore
> members of the power-set P(nodes) ) .
> And as Cantor showed , |P(nodes)| > |nodes|


All what Cantor showed concerns FISs. On the other hand he showed that
the set of FISs is countable.
>
> When you talk about an irrational number "in the binary tree" , you
> can't talk about it as 'any' FIS to represent it .


In fact there is no decimal representation of any irrational number.

> You have to use a whole sequence of FIS to represent it .

That can only be done by a finite definition - not by gathering
infinitely many FISs, obviously.

> Real numbers in general are fully represented only by 'infinite'
> paths , so they are uncountable .
> There are uncountable ways of combining your "finite-length
> paths" (that are countable in number) , by union, into "infinite-
> length paths" (that are uncountable in number) .


Not in the Binary Tree. All paths that are possible there (and are
represented by sets of nodes) belong to a countable set. It is
impossible to distinguish more than countably many. And it is
impossible to apply the diagonal argument to the Binary Tree, since it
contains all possible paths.
>
> You say that all paths are unions of the "finite paths" , and because
> the "finite paths" are countable , so are all paths .
> But that does not follow .


This construction of the Binary Tree shows that it is impossible to
discern uncountably many paths by nodes. (But Cantor's argument
concerns nodes (like d_n) only!)

The Binary Tree contains only those paths that are constructed step by
step of its initial segments Bk:

B0 = a0.

B1 = a0.
/
a1

B2 = a0.
/ \
a1 a2
...

Bk = a0.
/ \
a1 a2
...
/
... ak

...


Nothing else can be distinguished by nodes (like d_n).

Regards, WM



Date Subject Author
4/12/13
Read Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology § 246
JT
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology � 246
Virgil
4/12/13
Read Re: Matheology § 246
Ralf Bader
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology � 246
Virgil
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology � 246
Virgil
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology � 246
Virgil
4/13/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology � 246
Scott Berg
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology � 246
Virgil
4/13/13
Read Re: Matheology � 246
Scott Berg
4/13/13
Read Re: Matheology � 246
Virgil
4/13/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/13/13
Read Re: Matheology � 246
Virgil
4/13/13
Read Re: Matheology § 246
fom
4/13/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/13/13
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fom
4/13/13
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fom
4/13/13
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fom
4/13/13
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mueckenh@rz.fh-augsburg.de
4/13/13
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fom
4/13/13
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Virgil
4/13/13
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Virgil
4/13/13
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fom
4/12/13
Read Re: Matheology � 246
Virgil
4/12/13
Read Re: Matheology � 246
David C. Ullrich
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
dan.ms.chaos@gmail.com
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
fom
4/12/13
Read Re: Matheology § 246
Ralf Bader
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
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fom
4/12/13
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Virgil
4/13/13
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Ralf Bader
4/14/13
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mueckenh@rz.fh-augsburg.de
4/14/13
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Ralf Bader
4/14/13
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Virgil
4/12/13
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dan.ms.chaos@gmail.com
4/12/13
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mueckenh@rz.fh-augsburg.de
4/12/13
Read Re: Matheology § 246
dan.ms.chaos@gmail.com
4/12/13
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mueckenh@rz.fh-augsburg.de
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fom
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Virgil
4/12/13
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dan.ms.chaos@gmail.com
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mueckenh@rz.fh-augsburg.de
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Tanu R.
4/12/13
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dan.ms.chaos@gmail.com
4/13/13
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mueckenh@rz.fh-augsburg.de
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Tanu R.
4/13/13
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Tanu R.
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Tanu R.
4/13/13
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Tanu R.
4/13/13
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Virgil
4/13/13
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mueckenh@rz.fh-augsburg.de
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fom
4/13/13
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Virgil
4/13/13
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dan.ms.chaos@gmail.com
4/13/13
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mueckenh@rz.fh-augsburg.de
4/13/13
Read Re: Matheology § 246
dan.ms.chaos@gmail.com
4/13/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
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dan.ms.chaos@gmail.com
4/13/13
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mueckenh@rz.fh-augsburg.de
4/13/13
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Virgil
4/14/13
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mueckenh@rz.fh-augsburg.de
4/14/13
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Virgil
4/13/13
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Virgil
4/13/13
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Bergholt Stuttley Johnson
4/13/13
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Virgil
4/14/13
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mueckenh@rz.fh-augsburg.de
4/14/13
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Tanu R.
4/14/13
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Virgil
4/12/13
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Virgil
4/12/13
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Virgil
4/13/13
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fom
4/12/13
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Virgil
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Virgil
4/13/13
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mueckenh@rz.fh-augsburg.de
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Tanu R.
4/13/13
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Virgil
4/13/13
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David C. Ullrich
4/13/13
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mueckenh@rz.fh-augsburg.de
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fom
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mueckenh@rz.fh-augsburg.de
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fom
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Virgil
4/13/13
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fom
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David C. Ullrich
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mueckenh@rz.fh-augsburg.de
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dan.ms.chaos@gmail.com
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mueckenh@rz.fh-augsburg.de
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Virgil
4/14/13
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mueckenh@rz.fh-augsburg.de
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Tanu R.
4/14/13
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Virgil
4/13/13
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Virgil
4/14/13
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David C. Ullrich
4/14/13
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mueckenh@rz.fh-augsburg.de
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fom
4/14/13
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mueckenh@rz.fh-augsburg.de
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fom
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mueckenh@rz.fh-augsburg.de
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Virgil
4/15/13
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fom
4/16/13
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mueckenh@rz.fh-augsburg.de
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Bergholt Stuttley Johnson
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Virgil
4/14/13
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Virgil
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Virgil
4/15/13
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Virgil
4/16/13
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Virgil
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fom
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fom
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Tanu R.
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Virgil
4/17/13
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fom
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fom
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fom
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Virgil
4/17/13
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fom
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fom
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fom
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Virgil
4/17/13
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mueckenh@rz.fh-augsburg.de
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Virgil
4/17/13
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mueckenh@rz.fh-augsburg.de
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fom
4/17/13
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Virgil
4/17/13
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Tanu R.
4/17/13
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gus gassmann
4/18/13
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mueckenh@rz.fh-augsburg.de
4/18/13
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fom
4/21/13
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gus gassmann
4/21/13
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4/21/13
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fom
4/22/13
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mueckenh@rz.fh-augsburg.de
4/22/13
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Ralf Bader
4/22/13
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mueckenh@rz.fh-augsburg.de
4/22/13
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Bergholt Stuttley Johnson
4/23/13
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Ralf Bader
4/24/13
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mueckenh@rz.fh-augsburg.de
4/24/13
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Virgil
4/24/13
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Ralf Bader
4/25/13
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mueckenh@rz.fh-augsburg.de
4/25/13
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fom
4/25/13
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Ralf Bader
4/25/13
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mueckenh@rz.fh-augsburg.de
4/25/13
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Ralf Bader
4/25/13
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Tanu R.
4/25/13
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Ralf Bader
4/25/13
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Tanu R.
4/26/13
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Bergholt Stuttley Johnson
4/25/13
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Virgil
4/22/13
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mueckenh@rz.fh-augsburg.de
4/22/13
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fom
4/22/13
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gus gassmann
4/22/13
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gus gassmann
4/22/13
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mueckenh@rz.fh-augsburg.de
4/22/13
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fom
4/22/13
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mueckenh@rz.fh-augsburg.de
4/22/13
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fom
4/22/13
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Virgil
4/22/13
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Virgil
4/23/13
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mueckenh@rz.fh-augsburg.de
4/23/13
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fom
4/23/13
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mueckenh@rz.fh-augsburg.de
4/23/13
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fom
4/23/13
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Virgil
4/24/13
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mueckenh@rz.fh-augsburg.de
4/24/13
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Virgil
4/23/13
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fom
4/24/13
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mueckenh@rz.fh-augsburg.de
4/24/13
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Virgil
4/24/13
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fom
4/25/13
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Virgil
4/23/13
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Virgil
4/22/13
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4/18/13
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Virgil
4/19/13
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mueckenh@rz.fh-augsburg.de
4/19/13
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fom
4/19/13
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fom
4/19/13
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Virgil
4/19/13
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mueckenh@rz.fh-augsburg.de
4/19/13
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fom
4/19/13
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fom
4/19/13
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Virgil
4/18/13
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mueckenh@rz.fh-augsburg.de
4/18/13
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fom
4/18/13
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Virgil
4/18/13
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fom
4/17/13
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fom
4/17/13
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mueckenh@rz.fh-augsburg.de
4/17/13
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Virgil
4/17/13
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mueckenh@rz.fh-augsburg.de
4/17/13
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fom
4/17/13
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Virgil
4/17/13
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Bergholt Stuttley Johnson
4/17/13
Read Re: Matheology § 246
Ralf Bader
4/17/13
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fom
4/17/13
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Ralf Bader
4/17/13
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Tanu R.
4/18/13
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fom
4/17/13
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fom
4/17/13
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Virgil
4/17/13
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mueckenh@rz.fh-augsburg.de
4/17/13
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Virgil
4/17/13
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fom
4/15/13
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fom
4/13/13
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Virgil
4/13/13
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Tanu R.
4/13/13
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Tanu R.
4/13/13
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Tanu R.
4/13/13
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Virgil
4/13/13
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Virgil
4/13/13
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mueckenh@rz.fh-augsburg.de
4/13/13
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Virgil
4/14/13
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mueckenh@rz.fh-augsburg.de
4/14/13
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Tanu R.
4/14/13
Read Re: Matheology � 246
Virgil
4/13/13
Read Re: Matheology ? 246
Virgil

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