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

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dan.ms.chaos@gmail.com

Posts: 409
Registered: 3/1/08
Re: Matheology § 246
Posted: Apr 12, 2013 5:29 PM
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> 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 .


> However, since every FIS is the union of itself with all its
> predecessors, this sequence is also a sequence of unions, and, since
> it is not finite, contains unions of more than n FIS (for every n you
> can take). Usually this is called infinite uninon. So, if the path
> 0.111... is the union of all its FIS, then it should also be in list,
> since the list contains all finite unions, i.e., an infinite union.
> (Compare the union of all finite natural numbers results in an
> infinite set.)
>
> So we obtain a contradiction. 0.111... is in the list, if the list is
> understood as sequence of unions like
> {1} U {1, 2} U {1, 2, 3} U ... = |N,
> and it is not in the list, if the list is understood as sequence in
> mathematics.
>
> 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|

When you talk about an irrational number "in the binary tree" , you
can't talk about it as 'any' FIS to represent it .
You have to use a whole sequence of FIS to represent it . That's
called A PATH . So you need to use UNCOUNTABLE PATHS to represent
your UNCOUNTABLY MANY IRRATIONAL NUMBERS .

To recap :
Your tree has :
-Countable nodes :
eg. 0.11 , 0.1110 , 0101 ....
-Countable 'finite' paths : (paths that are made out of finite amount
of nodes)
eg. {0->0.1 -> 0.10 -> 0.101 -> 0.1011 }
UNCOUNTABLE 'infinite' paths: (paths that are made out of infinite
amount of nodes)
eg. {0->0.1 -> 0.11 -> 0.111 -> 0.1111 -> 0.11111 -> ......} to
represent the number "0.11111....."

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) .

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 .











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
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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
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Virgil
4/13/13
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fom
4/12/13
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Scott Berg
4/12/13
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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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Scott Berg
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
4/13/13
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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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fom
4/13/13
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fom
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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Virgil
4/13/13
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fom
4/12/13
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Virgil
4/12/13
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David C. Ullrich
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
4/12/13
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dan.ms.chaos@gmail.com
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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Ralf Bader
4/12/13
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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
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
dan.ms.chaos@gmail.com
4/12/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
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fom
4/12/13
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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
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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.
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Tanu R.
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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
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dan.ms.chaos@gmail.com
4/13/13
Read Re: Matheology § 246
mueckenh@rz.fh-augsburg.de
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
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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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4/12/13
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Virgil
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mueckenh@rz.fh-augsburg.de
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Tanu R.
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Virgil
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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
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fom
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David C. Ullrich
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mueckenh@rz.fh-augsburg.de
4/13/13
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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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Virgil
4/14/13
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mueckenh@rz.fh-augsburg.de
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Tanu R.
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Virgil
4/13/13
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Virgil
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David C. Ullrich
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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fom
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mueckenh@rz.fh-augsburg.de
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Virgil
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fom
4/16/13
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mueckenh@rz.fh-augsburg.de
4/16/13
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Bergholt Stuttley Johnson
4/16/13
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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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mueckenh@rz.fh-augsburg.de
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Virgil
4/16/13
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Virgil
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fom
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Virgil
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fom
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fom
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Virgil
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fom
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Virgil
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fom
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Virgil
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4/17/13
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gus gassmann
4/18/13
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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
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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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4/23/13
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4/22/13
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4/18/13
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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
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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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