Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Matheology § 191
Replies: 3   Last Post: Jan 12, 2013 6:15 PM

 Messages: [ Previous | Next ]
 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Matheology § 191
Posted: Jan 12, 2013 4:01 PM

On 12 Jan., 20:34, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 12 Jan., 16:24, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
> >> WM <mueck...@rz.fh-augsburg.de> writes:
> >> > Matheology § 191
>
> >> > The complete infinite Binary Tree can be constructed by first
> >> > constructing all aleph_0 finite paths and then appending to each path
> >> > all aleph_0 finiteley definable tails from 000... to 111...

>
> >> >         0
> >> >       1, 2
> >> >   3, 4, 5, 6
> >> > 7, ...

>
> >> > This Binary Tree contains aleph_0 * aleph_0 = aleph_0 paths.
>
> >> No, it contains aleph_0 * whatever the cardinality of the set of tails
> >> is.  Talk about begging the question!

>
> > A tail can be defined by a finite word *only*. Nobody can quote an
> > infinite string digit by digit - although most mathematicians believe
> > that instinctively when pondering about set theory (but never when
> > doing analysis).

>
> That's the part that was missing.  Without it you were begging the
> question because the set of paths and the set of tails are equinumerous.
> With it, the argument about tails and finite paths is pointless -- it's
> just padding around the same tired old claim.

It's only in order to easen the problem for you and others, who are
comprehending very slowly.

Paths are solely defined by nodes, according to set theory. (Otherwise
you could never gather uncountably many.)
Now try to find a path that is missing in the Binary Tree constructed
by countably many paths.
You fail.
But you insist in believing that old nonsense because it is so
familiar.
So you ask for further paths by means of finite definitions like 1/3.
Therefore I have appended all possible tails to all finite paths
(which would be sufficient to cover the tree but not sufficient to