Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set which is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.
A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number.
If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset.
If a set of sets is infinite or contains an infinite element, then its union is infinite. The powerset of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and a nonempty set is infinite. The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite; if the axiom of choice holds, then it is infinite.
If an infinite set is a well-ordered set, then it must have a nonempty subset which has no greatest element.
In ZF, a set is infinite if and only if the powerset of its powerset is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, infinite sets are precisely the Dedekind-infinite sets.
If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
Read more about this topic: Infinite Set
Famous quotes containing the word properties:
“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)
“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)