On 11 Dez., 19:53, Marshall <marshall.spi...@gmail.com> wrote: > 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.
No. Why should it? Of course the magnitude of natural numbers is not limited. For every number with n digits 333...333 there is another number with n^n digits. Nevertheless each one is finite.
> 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.
Of course. That is because there is no digit 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.
They *are* just the same, because your argument that above procedure would prove an infinite string of 3's is wrong. There is neither a natural nor a rational with an infinite string of digits. To be able to determine every digit of a number you like does not imply that there is a number with a never ending sequence of digits.
Why should it??? Because Cantor believed that God knows infinite strings? (He read it in civitate dei of St. Augustinus.)
Or because Zermelo misunderstood Bolzana-Dedekind's definition of infinity? Are you really thinking that infinity comes into being because a Dr. Zermelo of Germany said so?