In mathematics, even and odd ordinals extend the concept of parity from the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

s to the ordinal numbers. They are useful in some

transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...

proofs Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

. The literature contains a few equivalent definitions of the parity of an ordinal α: *Every

limit ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...

(including 0) is even. The

successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...

of an even ordinal is odd, and vice versa. *Let α = λ + ''n'', where λ is a limit ordinal and ''n'' is a natural number. The parity of α is the parity of ''n''. *Let ''n'' be the finite term of the

Cantor normal form In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...

of α. The parity of α is the parity of ''n''. *Let α = ωβ + ''n'', where ''n'' is a natural number. The parity of α is the parity of ''n''. *If α = 2β, then α is even. Otherwise α = 2β + 1 and α is odd. Unlike the case of even

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

s, one cannot go on to characterize even ordinals as ordinal numbers of the form

Ordinal multiplication In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an e ...

is not commutative, so in general In fact, the even ordinal cannot be expressed as β + β, and the ordinal number :(ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3 is not even. A simple application of ordinal parity is the

idempotence Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

law for cardinal addition (given the

well-ordering theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the orde ...

). Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum

References

{{reflist Ordinal numbers Parity (mathematics)