# proof that $\omega$ has the tree property

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 $T$.

Title proof that $\omega$ has the tree property ProofThatomegaHasTheTreeProperty 2013-03-22 12:52:36 2013-03-22 12:52:36 Henry (455) Henry (455) 5 Henry (455) Proof msc 05C05 msc 03E05 proof that omega has the tree property proof that infinity has the tree property