# 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^{} $X=\u27e8{x}_{0},{x}_{1},\mathrm{\dots}\u27e9$. 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 |
---|---|

Canonical name | ProofThatomegaHasTheTreeProperty |

Date of creation | 2013-03-22 12:52:36 |

Last modified on | 2013-03-22 12:52:36 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 5 |

Author | Henry (455) |

Entry type | Proof |

Classification | msc 05C05 |

Classification | msc 03E05 |

Synonym | proof that omega has the tree property |

Synonym | proof that infinity^{} has the tree property |