paradox of the binary tree

The complete infinite binary tree is a tree that consists of nodes (namely the numerals 0 and 1) such that every node has two children which are not children of any other node. The tree serves as binary representation of all real numbers of the interval [0,1] in form of paths, i.e., sequences of nodes.

Every finite binary treeMathworldPlanetmath with more than one level contains less paths than nodes. Up to level n there are 2^n paths and 2^(n+1) - 1 nodes.

Every finite binary tree can be represented as an ordered set of nodes, enumerated by natural numbersMathworldPlanetmath. The union of all finite binary trees is then identical with the infiniteMathworldPlanetmath binary tree. The paradoxMathworldPlanetmath is that, while the set of nodes remains countableMathworldPlanetmath as is the set of paths of all finite trees, the set of paths in the infinite tree is uncountable by Cantor’s theorem. (On the other hand, the paths are separated by the nodes. As no path can separate itself from another path without a node, the number of separated paths is the number of nodes.)

Literature: W. Mückenheim: Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006.

Title paradox of the binary tree
Canonical name ParadoxOfTheBinaryTree
Date of creation 2013-03-22 17:02:37
Last modified on 2013-03-22 17:02:37
Owner WM (16977)
Last modified by WM (16977)
Numerical id 21
Author WM (16977)
Entry type Definition
Classification msc 03E75
Classification msc 03E15
Synonym binary tree paradox
Defines complete binary treeMathworldPlanetmath
Defines complete infinite binary tree