On Thursday, January 31, 2013 8:55:12 AM UTC-8, WM wrote: > On 31 Jan., 15:27, forbisga...@gmail.com wrote: > > > > > The rational 1/3 > > > doesn't have a finite decimal expansion. None the less it is distinguishable > > > from every rational other than 1/3 at some place in the expansion > > > and only the infinite expansion can be calculated to be 1/3. > > > > If you are so sure about that than you should be able to find out > > whether 1/3 is among the paths that I have used to construct the > > complete infinite Binary Tree. Does it in fact differ from the union > > of all its finite initial segments > > 0.01 > > 0.0101 > > 0.010101 > > ... >
Yes it does, under the assumption that the continuation of all of your initial segments is not the repeating sequence . If it is then I will give you the repeating sequence 0. and it will not be among your finite initial segments unless you change your continuation at which point 0. will not be in your set. The alternative is you allow an infinite set of continuations for every finite initial segement and we are back to the point that you have no reason to claim the continuations countable.
When you multiply 0. by 100 you get 01. I can subtract 0. from that and get 1 then divide both size of the equation to get x = 1/11 (base 2) = 1/3 (base 10). None of your finite initial segements will work because you can't eliminate the fractional part that easily. (0.01 * 100) = 1 not 1.