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paradox of the binary tree

Defines: 
complete binary tree, complete infinite binary tree
Keywords: 
set theory, Cantor's theorem, uncountability
Synonym: 
binary tree paradox
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

03E75 no label found03E15 no label found

Comments

This article should detail the argument (more like it is in "W. Mückenheim: Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006.") In that paper the author makes a number of elementary errors that can be pointed out as being in error. In this short version, its not even clear what is being claimed, other than that in finite binary trees, you have one set of facts, and in the infinite case things change. Well, yeah, of course.

The claim "the number of separated paths is the number of nodes" may be an error if it were clearer what "separated paths" really meant. Every "separation" (i.e., every node node in fact) gives rise to an uncountable number of infinite paths below it that share that node and the path to the root as "prefixes". So what? For any finite length binary fractional number (.e.g, .00101010101) there are an uncountable number of binary numbers that start out that way. If "separated paths" is just a path, then clearly "the number of separated paths is the number of nodes" is out and out wrong.

Joe N

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