Re: Cantor and the binary tree

In article <1117022046.200086.282790@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter wrote:
> > In article <1116958479.555107.284630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> > > *** T. Winter wrote:
> > > > In article <1116939502.814879.192170@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
> > > mueckenh@xxxxxxxxxxxxxxxxx writes:
> > > > > If we accept that, in binary digits, SUM{n = 1 ... oo} 2^-n =
> > > > > 0.111... = 1
> > > >
> > > > You may note that that is *not* an infinite sum...
> >
> > Note what I said here. It is not a sum of infinitely many terms.
....
> > > Any path is an infinite sequence of bits which by multiplying with 2^-n
> > > and summing up establishes an infinite series representing a real
> > > number. Every combination of countably many bits is realized by
> > > definition.
> >
> > Mathematics does *not* define summing up infinitely many terms. It uses
> > limits in this case.
>
> Does mathematics allow me to write

Mathematics allows you a lot of things. Even saying 2 = 0. But it
does *not* definine a lot of things, like adding infinitely many terms.

> > > Why should 0.010101... not exist in that tree? Every path is infinite
> > > by definition as is 0.010101..., by definition.
> >
> > The path does exist, but there is no node at the end. Or if you wish
> > there is a node at the and (as J.H. Conway does with his surreal numbers),
> > the number of nodes is uncountable.
>
> There can be no node at the end, because there is no end. We do not
> need any node at the end in order to show that the set of nodes is
> equivalent to that of paths. It is shown by the following steps. Please
> point out which step is wrong.
>
> 1) Each real number of (0,1) is given by a path stretching over
> infinitely many nodes (bits).

Yup.

> 2) All nodes (bits) of the tree belong to a countable set.

Yup.

> 3) A node can only exist within a path.

Yup.

> 4) Any node increases the number of paths by 1 from 1 coming in, to 2
> going out. 2 - 1 = 1.

Eh? Now you are talking about an ever increasing finite tree, not about
an infinite tree. If you are talking about an infinite tree it may be
allowable, but because the number of paths is infinite, so it stays the
same when you add 1 to it.

> 5) Any node increases the number of nodes by 1.

Same remark here.

So what does that prove about infinite trees?
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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