On Dec 11, 9:13 am, WM <mueck...@rz.fh-augsburg.de> wrote: > On 11 Dez., 03:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > > > > > > If there are no infinite paths in that tree, 1/3 is not in that tree. > > > > > > 1/3 does not exist as a path. But everything you can ask for will be > > > found in the tree. > > > Everything of that kind is in the tree. > > > This makes no sense. Every path in the tree (if all paths are finite) is > > a rational with a power of 2 as the denominator. So 1/3 does not exist > > as a path. In what way does it exist in the tree? > > It exists in that fundamentally arithmetical way: You can find every > bit of it in my binary tree constructed from finite paths only. You > will fail to point to a digit of 1/3 that is missing in my tree. > Therefore I claim that every number that exists is in the tree.
This argument can be inverted to "prove" the existence of a natural number whose decimal expansion is an infinite string of 3s. The infinite-3 number exists in a fundamentally arithmetical way: you can find every digit of it in a preceding natural number constructed from finite string of 3s only. You will fail to point to a digit of ...333 that is missing in the natural numbers. Therefore you claim the naturals and the reals are just the same, but in the reverse direction, just as AP says they are.