The cardinality of the set of all root paths in the infinite complete binary tree is equal to the cardinality of the Continuum. The same holds true for kary trees for any finite k. But what is the case for k infinite?

Assuming your path has countable length, the set of all paths in a $k$ary tree will have cardinality $k^{\aleph_0}$. Indeed, at each step you have $k$ choices, and there are $\aleph_0$ steps (think of a path as a function from $\mathbb{N}$ to $[k]$). 


You may be also interested in the following paper by Shelah: http://shelah.logic.at/files/589.pdf In this paper (section 2) he defines the more general notion of the "tree revised power" of two cardinals k1, k2 as the supremum on the number of k2branches of trees with k1 nodes. He then proves that certain inequalities involving the tree revised power have some interesting consequences in pcf theory. 


protected by Scott Morrison♦ Jun 28 '13 at 16:28
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