```Date: Dec 18, 2009 9:19 AM
Author: Dik T. Winter
Subject: Re: Another AC anomaly?

In article <2cd3c08a-a072-4ef5-bc0b-f0aaa9126ea9@a21g2000yqc.googlegroups.com> WM <mueckenh@rz.fh-augsburg.de> writes: > On 17 Dez., 15:08, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article <6a57309a-a136-430c-a718-e38518c65...@q16g2000vbc.googlegroups= > .com> WM <mueck...@rz.fh-augsburg.de> writes: > >  > On 16 Dez., 03:44, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > >  > >  > ({1} U {1, 2} U {1, 2, 3} U ... = ({1} U {1, 2} U {1, 2, 3} U > >  > >  > U {1, 2, 3, ...} > >  > > >  > That is a matter of taste. > > > > No, it is a matter of convention.  In mathematics > >     a, b, c, ..., z > > means a, b, c, continue this way until you reach z.  But starting > >     {1}, {1, 2}, {1, 2, 3} > > and going on you never reach > >     {1, 2, 3, ...} >  > That is true. Therefore it does not exist.That you can not get there step by step does not mean that it does notexist. >                                            However, see Cantor, > collected works, p 445: > 0, 1, 2, 3, ... w_0, w_0 + 1, ..., gamma, ..., > He seems to reach far more.Right, he uses a convention that is no longer used. > >  > > A sequence of paths is not a path. > >  > > >  > But a union of paths is. > > > > No.  Suppose we have the paths 0.000 and 0.100, what is their union? And > > is it a path? >  > No. But the union contains two paths.Wrong.  If we look at the paths as sets, they are sets of nodes.  Theirunion is a set of nodes, not a set of paths.  And as a set of nodes wecan form from them seven different paths. >                                       And an infinite union of that > kind may contain the path 0.111...The union will contain a set of nodes.  With the nodes we can form paths(but they are not elements of the union).  But by your statements aninfinite sequence is not a path, and so 0.111... is not a path. > >  > > Here, again, you err.  You can not construct something in aleph_0 > >  > > steps; you will never complete your construction.  You *cannot* > >  > > get at aleph_0 > >  > > step by step. > >  > > >  > But you can make a bijection with all elements of omega? > > > > Yes, but not with a step by step method that will ever be complete. >  > The construction of the tree can be done within one step. Define: Let > there be every finite path. And every finite path is. The construction > of a path does not depend on a preceding step.It was in the construction *you* made.  But what is the definition of pathin this definition of the tree? > >  > >  > This hold for every limit of every sequence of finite paths. > >  > > > >  > > A limit is not a step by step process. > >  > > >  > Then assume it is a mapping from omega. > > > > Which mapping? >  > Let every finite path of every infinite path be mapped on the elements > of omega. That was simple.By your statements infinite paths do not exist.  But pray give such amapping.  Until now you have only asserted that such a mapping existswithout showing that.-- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/
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