proof that ω has the tree property

Let T be a tree with finite levels and an infiniteMathworldPlanetmath number of elements. Then consider the elements of T0. T can be partitioned into the set of descendants of each of these elements, and since any finite partitionMathworldPlanetmathPlanetmath of an infinite set has at least one infinite partition, some element x0 in T0 has an infinite number of descendants. The same procedure can be applied to the children of x0 to give an element x1T1 which has an infinite number of descendants, and then to the children of x1, and so on. This gives a sequenceMathworldPlanetmath X=x0,x1,. The sequence is infinite since each element has an infinite number of descendants, and since xi+1 is always of child of xi, X is a branch, and therefore an infinite branch of T.

Title proof that ω 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 infinityMathworldPlanetmath has the tree property