Even and odd ordinals
Encyclopedia
In mathematics
, even and odd ordinals extend the concept of parity from the natural number
s to the ordinal number
s. They are useful in some transfinite induction
proofs.
The literature contains a few equivalent definitions of the parity of an ordinal α:
Unlike the case of even integer
s, one cannot go on to characterize even ordinals as ordinal numbers of the form Ordinal multiplication is not commutative, so in general In fact, the even ordinal cannot be expressed as β + β, and the ordinal number2 = (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
law for cardinal addition (given the well-ordering theorem
). 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, even and odd ordinals extend the concept of parity from the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s to the ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
s. They are useful in some transfinite induction
Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for instance to sets of ordinal numbers or cardinal numbers.- Transfinite induction :Let P be a property defined for all ordinals α...
proofs.
The literature contains a few equivalent definitions of the parity of an ordinal α:
- Every limit ordinal (including 0) is even. The successorSuccessor ordinalIn set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal...
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 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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, one cannot go on to characterize even ordinals as ordinal numbers of the form Ordinal multiplication is not commutative, so in general In fact, the even ordinal cannot be expressed as β + β, and the ordinal number2 = (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
Idempotence is the property of certain operations in mathematics and computer science, that they can be applied multiple times without changing the result beyond the initial application...
law for cardinal addition (given the well-ordering theorem
Well-ordering theorem
In mathematics, the well-ordering 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 ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
). 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