In article <10c118d2-4751-4fd9-ade8-30c2f26afe7f@i31g2000yqm.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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) > > _________________ > B_0 = > > K_0 > _________________ > B_1 = > > K_0 > / > K_1 > _________________ > B_2 = > > K_0 > / \ > K_1 K_2 > _________________ > B_3 = > > K_0 > / \ > K_1 K_2 > / > K_3 > _________________ > > B_4 = > > K_0 > / \ > K_1 K_2 > / \ > K_3 K_4 > _________________ > ... > _________________ > B_j = > > K_0 > / \ > K_1 K_2 > / \ > K_3 K_4 ... > ... > ... K_j > _________________ > ... > _________________ > > 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.
Note than in FOL+ZFC, the set N necessarily exists, and, with suitable definition of left-child and right-child is already a complete infinite binary tree with suitable subsets denoting paths.
E.g., for n = {0,1,2,3,...} let the left child of n be 2*n+1 and the right child of n be 2*n+2 Then one has immediately a complete infinite binary tree with infinite paths. E.g., the path having only left branchings is {2^n-1: n in N}
So that what WM claims abut binary trees does not hold in FOL+ZFC, nor in any system in which a set like N exists.
