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

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Virgil

Posts: 870
Registered: 7/27/09
Re: Another AC anomaly?
Posted: Dec 16, 2009 1:35 PM
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In article
<ba9f2c85-caa2-491d-9263-bea1bed3ab1e@p19g2000vbq.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 16 Dez., 03:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>

> >  >
> >  > Why that? Group, ring and field are treated in my lessons.
> >
> > You think that something that satisfies the ZF axioms being a collection of
> > sets is rubbish, while something that satisfies the ring axioms being a
> > ring is not rubbish?

>
> Yes, exactly that is true.


That WM thinks it true is enough to make serious mathematicians doubt it.
>
> >  > 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.
> >
> > Right.  But there is no finite path that contains them all.  I believe in
> > a path that contains them all, and that is an infinite path.

>
> There is a finite path that contains larger numbers than you can ever
> think of.


It is certainly true that an infinite path must contain such numbers,
but one can always think of the last number of a finite increasing
sequence.

> >
> >  >       Every digit that is on the diagonal of Canbtor's list is a
> >  > member of a finite initial segment of a real number.
> >
> > Right, but there is no finite initial segment that contains them all.

>
> That is pure opinion


it is certainly a better opinion than WM's contrary opinion. Which is
all WM has in refutation.



> >
> >  > You can only argue about such digits. And all of them (in form of
> >  > bits) are present in my binary tree.
> >
> > Right, but your tree does not contain infinite paths, as you explicitly
> > stated.

>
> The tree contains all paths that can be constructed by nodes, using
> the axiom of infinity. Which one would be missing?


All uncountably many endless ones.
> >
> >  > >  > 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.


You will fail to point to a binary digit missing missing from {0,1} but
you will not find 1/3 among them.


> >  > Therefore I claim that every number that exists is in the tree.
> >
> > In that case you have a very strange notion of "existing in the tree".
> > Apparently you do *not* mean "existing as a path".  So when you say that
> > the number of (finite) paths is countable, I agree, but 1/3 is not included
> > in that, because it is not a path according to your statements.

>
> It is. I constructed a finite path from the root node to each other
> node. Then I appended an infinite tail.


As soon as you append such a tail, you no longer have a finite path.


> When the tree is completed,
> then the tail becomes invisible


If it is there, then it is visible to eyes which can see.

, because every sequenece of nodes has
> been constructed. But if you don't believe me, then look at the tree:
> You can see and admire every node and its connection to the root and
> the continuing paths downwards. So let me know what you think is
> missing.


Common sense, for one thing.
> >
> >  > > 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?
> >
> > Apparently not in your tree.  In your tree a path is a finite sequence of
> > nodes.

>
> The tree is the union of all paths. There is no end.


Your tree is nonsense.

Here is a sensible model of a complete infinite binary tree:

It starts with the set of naturals, N, with each natural being a node.
For each node, n in N, its child nodes are 2*n and 2*n+1.
A path in such a tree is a maximal subset containing 1 and for each n
containing either 2*n or 2*n+1. but not both.
> >
> >  >                                          Everey node of 1/3 (that you
> >  > can prove to belong to 1/3) is in the tree.
> >
> > Right, but there is no path that denotes 1/3.
> >
> >  > >  Similar for 'pi' and 'e'.
> >  >
> >  > Yes. Every digit is available on request.
> >
> > Right, but there is no path that denotes either 'pi' or 'e'.

>
> Which node is missing?


That question has no meaning, since in WM's trees there are no paths at
all and thus no real numbers.
> >
> >  > > 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):
> >
> > There is no proof needed.  Apparently there are real numbers in your tree
> > without being a path, because each path is finite (by your own definition).

>
> You are wrong. The paths in the complete tree aer unions of all finite
> paths.


Then they are not finite. Either paths in the complete tree are
infinite, possibly modeled as infinite sets of finite paths from finite
sub-trees, or WM's tree is incomplete. Or both.
> >
> >  >                                                         Every digit of
> >  > every real number that can be shown to exist exists in the tree.


Every 'digit' in a binary tree is in {0,1}. So what?
> >
> > But not every real number is represented in the tree by a path.

>
> Every real number is there, that can be represented by digits.


Not unless those digits are an infinite sequence of digits.
> >
> >  > 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?
> >
> > Yes, by your own admissions.  You state (explicitly) that every path is
> > finite and it is easy to prove that every number that is represented by
> > such a path is a rational number with a denominator that is a power of 2.
> > So there are apparently real numbers of which every digit is in the tree
> > that are not represented as a path, like 1/3.

>
> Even if I appended the tail 010101... ?


Then you no longer have your alleged but phony finite path but an
actually infinite path, which is what we have been saying all along.


Date Subject Author
11/23/09
Read Re: Another AC anomaly?
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12/2/09
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of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
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11/27/09
Read Re: Another AC anomaly?
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11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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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
11/28/09
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11/28/09
Read Re: Another AC anomaly?
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11/29/09
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11/29/09
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11/29/09
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11/30/09
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11/30/09
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11/30/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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11/30/09
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11/30/09
Read Re: Another AC anomaly?
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11/30/09
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Read Re: Another AC anomaly?
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12/19/09
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12/19/09
Read Re: Another AC anomaly?
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12/22/09
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12/22/09
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/7/09
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
Dik T. Winter
12/21/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/21/09
Read Re: Another AC anomaly?
Virgil
12/22/09
Read Re: Another AC anomaly?
Dik T. Winter
12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
Marshall
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
Ralf Bader
12/27/09
Read Re: Another AC anomaly?
Virgil
12/13/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/13/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
K_h
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/5/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/6/09
Read Re: Another AC anomaly?
Virgil
12/5/09
Read Re: Another AC anomaly?
Carsten Schultz
12/2/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
george
12/1/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
Virgil
11/27/09
Read Re: Another AC anomaly?
anonymous.rubbertube@yahoo.com
11/30/09
Read Re: Another AC anomaly?
William Hughes
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?
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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