example of tree (set theoretic)
The set 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 even though no particular node has that height.
A more interesting tree using defines if and for some . Then is the root, and all numbers which are not powers of another number are in . Then all squares (which are not also fourth powers) for , and so on.
To illustrate the concept of a cofinal branch, observe that for any limit ordinal we can construct a -tree which has no cofinal branches. We let and . The tree then has disjoint branches, each consisting of the set for some . No branch is cofinal, since each branch is capped at elements, but for any , there is a branch of height . Hence the supremum of the heights is .
|Title||example of tree (set theoretic)|
|Date of creation||2013-03-22 12:52:27|
|Last modified on||2013-03-22 12:52:27|
|Last modified by||uzeromay (4983)|