Infinite

A set is said to be infinite when it is not finite. That is, there is no natural number \(n\) for which we can enumerate the elements of the set from \(1\) to \(n\) without missing any.

An ordinal is called infinite when it is the order type of an infinite well-ordered set. The smallest infinite ordinal is \(\omega\). Every ordinal larger than it is infinite, and every ordinal smaller than it is finite.

Likewise, a cardinal is called infinite when it is the cardinality of an infinite set. The smallest infinite cardinal is \(\aleph_0\). Every cardinal larger than it is infinite, and every cardinal smaller than it is finite.

There are various equivalent ways to phrase the definition of an infinite set. Assuming the axiom of choice, a set \(S\) is infinite if and only if

• It is Dedekind infinite, that is, there is a strict subset \(T\subset S\) such that \(S\) and \(T\) are in bijection.
• It is in bijection with the disjoint union \(S\sqcup S\).
• It is in bijection with the Cartesian product \(S\times S\).

Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called amorphous sets.

Further kinds of infinite sets include countable and uncountable sets.

Properties

• Any superset of an infinite set is infinite. In particular, the union of an infinite set and any other set is infinite.
• The powerset of a infinite set is infinite.
• The set difference of an infinite set and a finite set is infinite.
• The sum, product, or exponentiation of an infinite ordinal with any other ordinal is infinite, except in the case of multiplying or exponentiating by 0, and exponentiation with base 0 or 1.
• The sum, product, or exponentiation of an infinite cardinal with any other cardinal is infinite, with the same exceptions as above.

External links

• Wikipedia Contributors. "Infinite set".

• Weisstein, Eric W. "Infinite Set" at MathWorld.