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

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ross.finlayson@gmail.com

Posts: 2,713
Registered: 2/15/09
Re: Another AC anomaly?
Posted: Dec 19, 2009 6:21 PM
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On Dec 19, 12:14 pm, Virgil <Vir...@home.esc> wrote:
> In article
> <2fac8bb1-4c90-4421-b559-1ea7f0301...@e27g2000yqd.googlegroups.com>,
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 18 Dez., 15:12, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>
> > >  > > Aha, you are clearly a mindreader.  Well, as far as I know mindreading
> > >  > > is not part of mathematics.  Anyhow, I can think of numbers larger
> > > than
> > >  > > that path.
> > >  > > But that is completely irrelevant.  I am able to think about a set
> > > that
> > >  > > contains all natural numbers, you apparently are not.
> > >  >
> > >  > How do you know that without confirming it by thinking the last too?

>
> > > Why need I to think about a last one (which there isn't) to be able to
> > > think
> > > about a set that contains all natural numbers?  Apparently you have some
> > > knowledge about how my mind works that I do not have.

>
> > Yes. A very convincing and often required proof of completenes of a
> > linear set is to know the last element.

>
> Knowing the allegedly last element in a set does not work unless one has
> imposed a linear ordering on the set, which is in no wise necessary for
> sethood. Which is the "last" point of the set of points on a circle?
>
>
>

> > One of the rebuttals has meanwhile been changed. Peter Webb
> > recognized: It is true that you cannot show pi as a finite decimal,
> > but you can't show 1/3 as a finite decimal either.

>
> > Just what I said.
>
> > >  > >  > The tree contains all paths that can be constructed by nodes, using
> > >  > >  > the axiom of infinity. Which one would be missing?
> > >  > >
> > >  > > The infinite paths because you stated a priori that your tree did not
> > >  > > contain infinite paths.  So it is impossible to construct in your tree
> > >  > > infinite paths by the axiom of infinity.
> > >  >
> > >  > The axiom of infinity establishes the set N from finite numbers.

>
> > > It establishes the *existence* of a set N of finite numbers.
>
> > What else should be established?
>
> > >  > It establishes the infinite paths as well in my tree from finite
> > >  > paths.

>
> > > No.  That is impossible because you stated that the paths were finite.
> > > What it *does* establish is the extistence of a set P of finite paths.

>
> > It is rather silly to argue about the uncountability of the set of
> > paths.

>
> Then stop doing it. There is a perfectly adequate proof that one cannot
> have a list of paths that contains all paths, or , equivalently, one
> cannot have a list of all subset of N.
>  Since being capable of being listed is a necessary requirement for
> countability, that proof eliminates countability for the set of all
> paths s well as for the power set of any infinite set.
>

> > Only minds completely disformed by set theory could try to
> > defend the obviously false position that there were uncountably many
> > paths.

>
> Minds thus "deformed" by set theory are often capable of creating valid
> proofs that WM not only cannot create but also cannot even follow.
>
>
>

> > But "10 Questions" will give you the answer why there are not
> > uncountably many paths. There are no infinite decimal expansions of
> > real numbers. There are not due paths in the tree.

>
> 1 = 1.000... is a real number with an infinite decimal expansion, and in
> ZF there are infinitely many more of them.
>
> I know not what a "due path" is, but in my infinite binary tree, whose
> nodes are the members of N, every path is an infinite subset of N, and
> there are more of them than even WM can count.
>
>
>

> > It is as impossible to express any real number by an infinite decimal
> > expansion as it is impossible to express 0 by a unit fraction. The
> > rest will be explained in "10 Questions". Therefore I will stop with
> > this topic here.

>
> Then in WM's world, 1.000... is not a real number.
>
> Odd of him, but he has always been odd.


There you assume the consistency of the statement "Dedekind/Cauchy is
complete", where it's a constructive argument. (Union of countables.)

Otherwise there might be systems where that is not so yet
constructively irrelevant.

Ross Finlayson



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Read Re: Another AC anomaly?
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11/29/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
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11/30/09
Read Re: Another AC anomaly?
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11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
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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
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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11/29/09
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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?
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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?
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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?
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/11/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
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12/15/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
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12/16/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/17/09
Read Re: Another AC anomaly?
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12/17/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/4/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
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12/7/09
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12/7/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/7/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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12/8/09
Read Re: Another AC anomaly?
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12/10/09
Read Re: Another AC anomaly?
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12/10/09
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12/10/09
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12/10/09
Read Re: Another AC anomaly?
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12/11/09
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12/11/09
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12/11/09
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12/11/09
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12/11/09
Read Re: Another AC anomaly?
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12/11/09
Read Re: Another AC anomaly?
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12/12/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/18/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/19/09
Read Re: Another AC anomaly?
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12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
Virgil
12/19/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/22/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
Virgil
12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/27/09
Read Re: Another AC anomaly?
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12/13/09
Read Re: Another AC anomaly?
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12/13/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
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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?
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12/2/09
Read Re: Another AC anomaly?
Virgil
12/1/09
Read Re: Another AC anomaly?
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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?
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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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