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Re: Matheology § 191
Posted:
Jan 12, 2013 12:03 PM
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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).
Therefore every definition of a tail belongs to the set of finite
words and hence to a set of cardinality less than aleph_0. (In fact
during the lifetime of the universe the set of used words will remain
finite.)
Regards, WM
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