Finite

A set is said to be finite if its elements can be labeled with the numbers from \(1\) to \(n\), for some natural number \(n\). A set that isn't finite is said to be infinite.

More precisely, a set \(S\) is finite when there exists \(n\in \mathbb N\) and a bijective function (a one-to-one correspondence) \(f:S\to\{<n\}\), where \(\{<n\}\) denotes the set of naturals less than \(n\). In terms of von Neumann ordinals, this is equivalent to there being some well-ordering on the set whose order-type is finite. The unique natural for which this holds is called its cardinality, although this concept may be defined in greater generality. Perhaps the simplest finite set is the empty set \(\varnothing\), whose cardinality is 0. Any singleton set \(\{a\}\) is finite and has cardinality 1.

An ordinal is called finite when it's the order type of a finite well-ordered set. It can be proven that these ordinals correspond precisely to the order types of the sets \(\{<n\}\). This allows for the identification of finite ordinals with natural numbers.

Likewise, a cardinal is called finite when it's the cardinality of a finite set. Once again, finite cardinals can be identified with the natural numbers.

Dedekind finiteness

Dedekind has the following definition of finiteness that does not make reference to \(\mathbb N\):

\(X\) is finite if there is no proper subset of \(X\) that has the same cardinality as \(X\).

Without the axiom of choice it cannot be proven that this is equivalent to the \(\mathbb N\)-based definition of finiteness.^{[Citation needed]}

Properties

• Any subset of a finite set is finite. In particular, the intersection of a finite set and any other set is finite.
• The powerset of a finite set is finite.
• The union of two finite sets, and thus of finitely many finite sets, is finite.
• The sum, product, or exponentiation of two finite ordinals is finite.
• The sum, product, or exponentiation of two finite cardinals is finite.

External links

• Barile, Margherita. "Finite Set" at MathWorld.

• Wikipedia Contributors. "Finite set".