On Apr 16, 10:58 pm, calvin <cri...@windstream.net> wrote: > On Apr 16, 10:36 pm, Dave <dave_and_da...@Juno.com> wrote: > > > On Apr 16, 9:05 pm, calvin <cri...@windstream.net> wrote: > > > > Can one be haunted by a mathematical concept? > > > The binary tree answers yes for me. That I can > > > construct, in a countable number of steps, a fully > > > comprehensible representation of every real number > > > between zero and one, is astonishing every time > > > I think of it. > > > > What else in math is so easy to do and has equally > > > breathtaking results, I wonder. > > > I was with you until you said "every real number." Are you sure you > > don't mean "any real number"? > > All real numbers between zero and one. The construction > I have in mind is the simple one of starting with a > binary point at the topmost node, and then going from > left to right below it to the two nodes at that level, > and then going from left to right for the four nodes > at the next level, and so on. Of course it helps > visualization to halve the distance down to each level > so that the possible paths from node to node are seen > approaching individual points on the unit line at the > bottom, but my main point is that this construction > requires a countable number of steps to lay out the > uncountable number of paths to the real numbers between > zero and one.
Is this more astonishing than, say, one can write down any real number between 0 and 1 in a countable number of digits, even though for each digit, one has only 10 choices.