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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 7, 2009 11:20 AM
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On 7 Dez., 16:41, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> In article <5f8c7e7f-ec83-45ba-a584-b61475969...@j4g2000yqe.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
>  > On 3 Dez., 16:27, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
>  >
>  > >  > > No.  I asked you for a mathematical definition of "actual infinity"
>  > >  > > and you told me that it was "completed infinity".  Next I asked you
>  > >  > > for a mathematical definition of "completed infinity" but you have
>  > >  > > not given an answer.  So I still do not know what either "actual
>  > >  > > infinity" or "completed infinity" are.
>  > >  >
>  > >  > Both are nonsense. But both are asumed to make sense in set theory.
>  > >
>  > > No, set theory does not contain a definition of either of them.
>
> So in what way do they make sense in set theory?  As I have not seen a
> definition of them at all, I can not see what that means.


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.
>
>  > >  > The axiom of infinity is a definition of actual infinity.
>  > >  > "There *exists* a set such that ..."
>  > >  > Without that axiom there is only potential infinity, namely Peano
>  > >  > arithmetic.
>  > >
>  > > I see neither a definition of the words "actual infinity" neither
>  > > a definition of "potential infinity".  Or do you mean that "potential
>  > > infinity" is Peano arithmetic (your words seem to imply that)?
>  > >
>  > > So we can say that in "potential infinity" consists of a set of axioms?
>  >
>  > Here is, to my knowledge, the simplest possible explanation.
>
> Darn, I ask for a definition, not for an explanation.


The definition of an actually infinite set is given in set theory by
the axiom of infinity. You should know it or know where to find it.
(You can look it up in my book.)

The definition of a potentially infinite set is given by
1 in N
n in N then n+1 in N.
>
>  >                                                              Consider
>  > the infinite binary tree:
>  >
>  >     0
>  >     /\
>  >   0  1
>  >  /\    /\
>  > 0 1 0 1
>  > ...
>  >
>  > Paint all paths of the form
>  > 0.111...
>  > 0.0111...
>  > 0.00111...
>  > 0.000111...
>  > and so on.
>  > Potential infinity then says that every node and every edge on the
>  > outmost left part of the tree gets painted.
>  > Actual infinity says that there is a path 0.000... parts of which
>  > remain unpainted. And that is wrong.
>
> So "potential infinity" and "actual infinity" are theories?  But I do not
> know of any theory that states that there is any part of the path 0.000...
> that is unpainted.


That is the inconsistency of set theory.

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.

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.)

Regards, WM



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mueckenh@rz.fh-augsburg.de
11/27/09
Read Re: Another AC anomaly?
Virgil

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