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.)