# example of tree (set theoretic)

The set ${\mathbb{Z}}^{+}$ is a tree with $$. This isn’t a very interesting tree, since it simply consists of a line of nodes. However note that the height is $\omega $ even though no particular node has that height.

A more interesting tree using ${\mathbb{Z}}^{+}$ defines $$ if ${i}^{a}=m$ and ${i}^{b}=n$ for some $i,a,b\in {\mathbb{Z}}^{+}\cup \{0\}$. Then $1$ is the root, and all numbers which are not powers of another number are in ${T}_{1}$. Then all squares (which are not also fourth powers) for ${T}_{2}$, and so on.

To illustrate the concept of a cofinal branch, observe that for any limit ordinal^{} $\kappa $ we can construct a $\kappa $-tree which has no cofinal branches. We let $$ and $$. The tree then has $\kappa $ disjoint branches, each consisting of the set $$ for some $$. No branch is cofinal, since each branch is capped at $\beta $ elements, but for any $$, there is a branch of height $\gamma +1$. Hence the supremum of the heights is $\kappa $.

Title | example of tree (set theoretic) |
---|---|

Canonical name | ExampleOfTreesetTheoretic |

Date of creation | 2013-03-22 12:52:27 |

Last modified on | 2013-03-22 12:52:27 |

Owner | uzeromay (4983) |

Last modified by | uzeromay (4983) |

Numerical id | 5 |

Author | uzeromay (4983) |

Entry type | Example |

Classification | msc 05C05 |

Classification | msc 03E05 |

Related topic | CofinalBranch |