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. Any countable set is infinite, but there are non-countable infinite sets. An ordinal is called infinite when it is the order type of an infinite well-ordered set. Under the von Neumann representation, this is just equivalent to the ordinal itself being infinite, as a 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, and this is also equal to the cardinal itself being infinite under the definition of cardinals as initial ordinal. 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 \(S\times S\).
• \(S\) has a countable subset.

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

Within ZFC, the existence of infinite sets is guaranteed by the axiom of infinity, which implies the existence of \(\omega\), which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. A model of this theory is provided by the set of hereditarily finite 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.

Infinity

A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting limits to infinity, series, and improper integrals, or as an object in a structure such as the extended real numbers. However, there is no real number that serves the purpose of infinity, since the real numbers have the Archimedean property, meaning that for every real number \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\).

External links

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

• Weisstein, Eric W. "Infinity" at MathWorld.

• Wikipedia Contributors. "Infinite set".

• Wikipedia Contributors. "Infinity".