On 19 Jun., 11:07, "|-|ercules" <radgray...@yahoo.com> wrote:
> The list of computable reals contains every digit (in order) of all possible infinite sequences.
Hi Herc,
why not instead of a list of all reals produce a Binary Tree. This tree can be shown to produce every infinite binary sequence that can be produced by the following step-by-step construction. This construction is possible, because the set of all nodes is a countable set and all paths exist among the nodes and nowhere else. The construction is as follows:
The Binary Tree contains all real numbers of the interval [0, 1] as infinite paths.
0, / \ 0 1 / \ / \ 0 1 0 1 / 0 ...
The nodes K_i with numerical values 0 or 1 are countable:
K_0 / \ K_1 K_2 / \ / \ K_3 K_4 K_5 K_6 / K_7 ...
Everey step adds one node to the configuration B_i and yields configuration B_(i+1)
There is no end, hence there is no node that is not constructed. If there is no infinite path constructed at all, this means either that infinite paths consist not only of nodes (but of phantasy-products of matheologicians) or they do not exist at all.
The latter is true. There is no completed infinite path but there is merely the possibility to add a node to any path of any finite length. But that does not yield an uncountable set of paths.
