proof that has the tree property
Let be a tree with finite levels and an infinite number of elements. Then consider the elements of . 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 in has an infinite number of descendants. The same procedure can be applied to the children of to give an element which has an infinite number of descendants, and then to the children of , and so on. This gives a sequence . The sequence is infinite since each element has an infinite number of descendants, and since is always of child of , is a branch, and therefore an infinite branch of .
|Title||proof that has the tree property|
|Date of creation||2013-03-22 12:52:36|
|Last modified on||2013-03-22 12:52:36|
|Last modified by||Henry (455)|
|Synonym||proof that omega has the tree property|
|Synonym||proof that infinity has the tree property|