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

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Dik T. Winter

Posts: 7,899
Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 10, 2009 10:35 AM
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In article <fd82ac52-8f71-4930-8f8e-415187ae832b@g26g2000yqe.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes:
> On 8 Dez., 16:07, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>

> > > You have seen the axiom of infinity. It say that an infinite set
> > > exists and that implies that infinitely many elements of that set
> > > exist. That is actual infinity.

> >
> > Oh, so actual infinity means that a set with infinitely many elements
> > exists?

>
> Yes.
>

> > In that case you should reject the axiom of infinity. You are allowed
> > to do that, and you will get different mathematics. But you can not
> > claim that mathematics with the axiom of infinity is nonsense just
> > because you do not like it. But go ahead without the axiom of infinity,
> > I think you have to redo quite a bit of mathematics.

>
> Before 1908 there was quite a lot of mathematics possible.


Yes, and since than quite a lot of newer mathematics has been made available.
Moreover, before 1908 mathematicians did use concepts without actually
defining them, which is not so very good in my opinion.

> > > The definition of an actually infinite set is given in set theory by
> > > the axiom of infinity.

> >
> > You are wrong, the axiom of infinity says nothing about "actually
> > infinite set". Actually the axiom of infinity does not define anything.
> > It just states that a particular set with a particular property does
> > exist.

>
> That is just the definition of actual infinity.


I see no definition, so what *is* the definition?

> > > The definition of a potentially infinite set is given by
> > > 1 in N
> > > n in N then n+1 in N.

> >
> > That does not make sense. Without the axiom of infinity the set N does
> > not necessarily exist, so stating 1 in N is wrong unless you can prove
> > that N does exist or have some other means to have the existence of N,
> > but that would be equivalent to the axiom of infinity.

>
> N need not exist as a set. If n is a natural number, then n + 1 is a
> natural numbers too. Why should sets be needed?


Ok, so N is not a set. What is it?

> > > The complete infinite binary tree can be constructed using countably
> > > many finite paths (each one connecting a node to the root node), such
> > > that every node is there and no node is missing and every finite path
> > > is there and no finite path is missing.

> >
> > Right.
> >

> > > Nevertheless set theory says that there is something missing in a tree
> > > thus constructed. What do you think is missing? (If nothing is
> > > missing, there are only countably many paths.)

> >
> > And here again you are wrong. There are countably many finite paths.
> > There are not countably many infinite paths, and although you have tried
> > many times you never did show that there were countably many infinite
> > paths.

>
> There is not even one single infinite path!


Eh? So there are no infinite paths in that 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.
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.
--
dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/


Date Subject Author
11/23/09
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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?
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
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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11/28/09
Read Re: Another AC anomaly?
Virgil
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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Read Re: Another AC anomaly?
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11/30/09
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11/30/09
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11/29/09
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11/26/09
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11/26/09
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11/26/09
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11/29/09
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11/27/09
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11/27/09
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11/30/09
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11/27/09
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Read Re: Another AC anomaly?
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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?
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Read Re: Another AC anomaly?
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11/27/09
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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
11/30/09
Read Re: Another AC anomaly?
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12/1/09
Read Re: Another AC anomaly?
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Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
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mueckenh@rz.fh-augsburg.de
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Read Re: Another AC anomaly?
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12/21/09
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12/21/09
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12/22/09
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12/11/09
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12/11/09
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12/12/09
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12/10/09
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12/10/09
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12/11/09
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12/7/09
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12/8/09
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12/8/09
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12/8/09
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12/8/09
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12/8/09
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12/10/09
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12/3/09
Read Re: Another AC anomaly?
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12/3/09
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12/3/09
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12/7/09
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12/7/09
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12/7/09
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12/7/09
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12/8/09
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12/8/09
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12/8/09
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12/11/09
Read Re: Another AC anomaly?
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12/15/09
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12/16/09
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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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12/17/09
Read Re: Another AC anomaly?
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12/17/09
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12/4/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
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12/7/09
Read Re: Another AC anomaly?
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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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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
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12/12/09
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12/4/09
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12/4/09
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12/4/09
Read Re: Another AC anomaly?
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12/4/09
Read Re: Another AC anomaly?
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12/5/09
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12/1/09
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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
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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
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11/27/09
Read Re: Another AC anomaly?
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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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