Other Properties
If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n − 1, n − 2, ..., 2, 1 has length n for any n.
The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relation: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X,R) is isomorphic to (C,∈).
Read more about this topic: Well-founded Relation
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)