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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: Dec 11, 2009 12:13 PM
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On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <fec95b83-39c5-4537-8cf7-b426b1779...@k17g2000yqh.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
>  > On 10 Dez., 16:35, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>  > >  > Before 1908 there was quite a lot of mathematics possible.
>  > >
>  > > Yes, and since than quite a lot of newer mathematics has been made
>  > > available.
>  >
>  > Most of it being rubbish.
>
> Nothing more than opinion while you have no idea what has been done in
> mathematics since 1908.  Algebraic number theory is rubbish?


The answer is an explicit no. "Most" here concerns the magnitude of
numbers involved. There was much ado about inaccessible cardinals.
>
>  > > Moreover, before 1908 mathematicians did use concepts without actually
>  > > defining them, which is not so very good in my opinion.
>  >
>  > Cantor gave a definition of set. What is the present definition?
>
> Something that satisfies the axioms of ZF (when you are working within ZF).
> It is similar to the concepts of group, ring and field.  Something that
> satisfies those axioms is such a thing.  But I think you find all those
> things rubbish.


Why that? Group, ring and field are treated in my lessons.

>
>  > >  > There is not even one single infinite path!
>  > >
>  > > Eh?  So there are no infinite paths in that tree?
>  >
>  > In fact no, but every path that you believe in is also in the tree,
>  > i.e., you will not be able to miss a path in the tree.
>
> I believe in infinite paths, you state they are not in the tree.  So we
> have a direct contradiction to your assertion.


You believe in infinite paths. But you cannot name any digit that
underpins your belief. Every digit that you name belongs to a finite
path. Every digit that is on the diagonal of Canbtor's list is a
member of a finite initial segment of a real number.

You can only argue about such digits. And all of them (in form of
bits) are present in my binary tree.
>
>  > >  >                                   But there is every path
>  > >  > which you believe to be an infinite path!! Which one is missing in
>  > >  > your opinion? Do you see that 1/3 is there?
>  > >
>  > > If there are no infinite paths in that tree, 1/3 is not in that tree.
>  >
>  > 1/3 does not exist as a path. But everything you can ask for will be
>  > found in the tree.
>  > Everything of that kind is in the tree.
>
> This makes no sense.  Every path in the tree (if all paths are finite) is
> a rational with a power of 2 as the denominator.  So 1/3 does not exist
> as a path.  In what way does it exist in the tree?


It exists in that fundamentally arithmetical way: You can find every
bit of it in my binary tree constructed from finite paths only. You
will fail to point to a digit of 1/3 that is missing in my tree.
Therefore I claim that every number that exists is in the tree.

>  > > Otherwise 1/3 would be a rational with a denominator that is a power of
>  > > 2 (each finite path defines such a number).
>  > >
>  > >  > What node of pi is missing in the tree constructed by a countable
>  > >  > number of finite paths (not even as a limit but by the axiom of
>  > >  > infinity)?
>  > >
>  > > By the axiom of infinity there *are* infinite paths in that tree.  So your
>  > > statement that there are none is a direct contradiction of the axiom of
>  > > infinity.
>  >
>  > Try to find something that exists in your opinion but that does not
>  > exist in the tree that I constructed.
>
> In what way do numbers like 1/3 exist in your tree?  Not as a path, apparently,
> but as something else.


Isn't a path a sequence of nodes, is it? Everey node of 1/3 (that you
can prove to belong to 1/3) is in the tree.


>  Similar for 'pi' and 'e'.

Yes. Every digit is available on request.

> So when you state that
> the number of paths is countable that does not mean that the number of real
> numbers is countable because there are apparently real numbers in your tree
> without being a path.


Wrong. Not only "apparantly" but provably (on request): Every digit of
every real number that can be shown to exist exists in the tree.

Or would you say that a number, every existing digit of which can be
shown to exist in the tree too, is not in the tree as a path?

Regards, WM



Date Subject Author
11/23/09
Read Re: Another AC anomaly?
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12/1/09
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12/1/09
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12/2/09
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of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
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11/26/09
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11/28/09
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11/26/09
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mueckenh@rz.fh-augsburg.de
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11/28/09
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11/29/09
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mueckenh@rz.fh-augsburg.de
11/29/09
Read Re: Another AC anomaly?
Virgil
11/29/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
Marshall
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
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11/29/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/26/09
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11/26/09
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11/26/09
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11/29/09
Read Re: Another AC anomaly?
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11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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11/29/09
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11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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11/27/09
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/30/09
Read Re: Another AC anomaly?
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11/30/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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12/1/09
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Read Re: Another AC anomaly?
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12/8/09
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12/10/09
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Read Re: Another AC anomaly?
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12/16/09
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Read Re: Another AC anomaly?
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12/17/09
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Read Re: Another AC anomaly?
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12/7/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/4/09
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Read Re: Another AC anomaly?
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12/5/09
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12/5/09
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11/27/09
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11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
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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