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[parent] proof that $\omega$ has the tree property (Proof)

Let $ T$ be a tree with finite levels and an infinite number of elements. Then consider the elements of $ T_0$. $ T$ can be partitioned into the set of descendants of each of these elements, and since any finite partition of an infinite set has at least one infinite partition, some element $ x_0$ in $ T_0$ has an infinite number of descendants. The same procedure can be applied to the children of $ x_0$ to give an element $ x_1\in T_1$ which has an infinite number of descendants, and then to the children of $ x_1$, and so on. This gives a sequence $ X=\langle x_0, x_1,\ldots \rangle$. The sequence is infinite since each element has an infinite number of descendants, and since $ x_{i+1}$ is always of child of $ x_i$, $ X$ is a branch, and therefore an infinite branch of $ T$.



"proof that $\omega$ has the tree property" is owned by Henry.
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Other names:  proof that omega has the tree property, proof that infinity has the tree property

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Cross-references: branch, child, sequence, children, partition, descendants, infinite, levels, finite, tree

This is version 2 of proof that $\omega$ has the tree property, born on 2002-07-27, modified 2002-08-02.
Object id is 3218, canonical name is ProofThatOmegaHasTheTreeProperty.
Accessed 2960 times total.

Classification:
AMS MSC: 05C05 (Combinatorics :: Graph theory :: Trees)
 03E05 (Mathematical logic and foundations :: Set theory :: Other combinatorial set theory)
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