Properties
The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable.
If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point): the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.
Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).
Read more about this topic: Limit Ordinal
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)