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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 k-ary trees for any finite k. But what is the case for k infinite?

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I don't know what you mean by $n$ tending towards infinity, but if you take a rooted tree such that each vertex has $\kappa$ outward nodes for any $2 \leq \kappa \leq \aleph_0$, then there are continuum-many infinite paths. Indeed, you have at least as many paths as you do when $\kappa = 2$ and no more paths than the number of sequences formed by the elements of the (countable) vertex set. –  Pete L. Clark May 23 '10 at 4:42
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Better yet: this is true for $2 \leq \kappa \leq 2^{\aleph_0}$ since $(2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0}$. –  François G. Dorais May 23 '10 at 4:53

2 Answers 2

up vote 5 down vote accepted

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 $[k]$).

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Kevin, though the number of paths is indeed $k^{\aleph_0}$, this number can be (MUCH) larger than both $k$ and $2^{\aleph_0}$. for example, take as $k$ the $\omega$-th successor of $2^{\aleph_0}$. Then $k^{\aleph_0}$ is obviously larger than the continuum, and it is also larger than $k$ by König's lemma. –  Andres Caicedo May 23 '10 at 7:15
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Successor cardinal, not successor ordinal. –  Harry Altman May 23 '10 at 7:40
    
Also, thanks for the comments! –  Kevin Ventullo May 23 '10 at 8:05
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Andres, I think you mean König's Theorem, not lemma. There are two Königs, the father a set theorist, the son a graph theorist. But König's Lemma usually refers to the statement that every infinite finitely-branching tree has an infinite branch –  Joel David Hamkins May 23 '10 at 12:11
    

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 k2-branches of trees with k1 nodes. He then proves that certain inequalities involving the tree revised power have some interesting consequences in pcf theory.

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

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