Реттік арифметиканың кеңейтілген амалдары

Operations on ordinals that extend classical arithmetic
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 explicit well ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.

1=α · 0 = 0 · α = 0, and the zero product property holds: 1=α · β = 0 → α = 0 or 1=β = 0. The ordinal 1 is a multiplicative identity, 1=α · 1 = 1 · α = α. Multiplication is associative, 1=(α · β) · γ = α · (β · γ). Multiplication is strictly increasing and continuous in the right argument: (α < β and γ > 0) → γ·α < γ·β. Multiplication is not strictly increasing in the left argument, for example, 1 < 2 but 1=1 · ω = 2 · ω = ω. However, it is (non strictly) increasing, i. e. α ≤ β → α·γ ≤ β·γ. Multiplication of ordinals is not in general commutative. Specifically, a natural number greater than 1 never commutes with any infinite ordinal, and two infinite ordinals α and β commute if and only if 1=α^(m) = β^(n) for some nonzero natural numbers m and n. The relation "α commutes with β" is an equivalence relation on the ordinals greater than 1, and all equivalence classes are countably infinite. Distributivity holds, on the left: 1=α(β + γ) = αβ + αγ. However, the distributive law on the right 1=(β + γ)α = βα+γα is not generally true: 1=(1 + 1) · ω = 2 · ω = ω while 1=1 · ω + 1 · ω = ω + ω, which is different. There is a left cancellation law: If α > 0 and 1=α · β = α · γ, then 1=β = γ. Right cancellation does not work, e. g. 1=1 · ω = 2 · ω = ω, but 1 and 2 are different. A left division with remainder property holds: for all α and β, if β > 0, then there are unique γ and δ such that 1=α = β · γ + δ and 1=δ < β. Right division does not work: there is no α such that 1=α · ω ≤ ω^(ω) ≤ (α + 1) · ω. The ordinal numbers form a left near semiring, but do not form a ring. Hence the ordinals are not a Euclidean domain, since they are not even a ring; furthermore the Euclidean "norm" would be ordinal valued using the left division here. A δ number (see Multiplicatively indecomposable ordinal) is an ordinal β greater than 1 such that 1=αβ = β whenever 0 < α < β. These consist of the ordinal 2 and the ordinals of the form 1=β = ω^(ω^(γ)).

1=α^(0) = 1. If 0 < α, then 1=0^(α) = 0. 1=1^(α) = 1. 1=α^(1) = α. 1=α^(β)·α^(γ) = α^(β + γ). 1=(α^(β))^(γ) = α^(β·γ). There are α, β, and γ for which 1=(α · β)^(γ) ≠ α^(γ) · β^(γ). For instance, 1=(ω · 2)^(2) = ω·2·ω·2 = ω^(2) · 2 ≠ ω^(2) · 4. Ordinal exponentiation is strictly increasing and continuous in the right argument: If γ > 1 and α < β, then γ^(α) < γ^(β). If α < β, then α^(γ) ≤ β^(γ). Note, for instance, that 2 < 3 and yet 1=2^(ω) = 3^(ω) = ω. If α > 1 and 1=α^(β) = α^(γ), then 1=β = γ. If 1=α = 1 or 1=α = 0 this is not the case. For all α and β, if β > 1 and α > 0 then there exist unique γ, δ, and ρ such that 1=α = β^(γ) · δ + ρ such that 0 < δ < β and ρ < β^(γ). Jacobsthal showed that the only solutions of 1=α^(β) = β^(α) with α ≤ β are given by 1=α = β, or 1=α = 2 and 1=β = 4, or α is any limit ordinal and 1=β = εα where ε is an ε number larger than α.

There are ordinal operations that continue the sequence begun by addition, multiplication, and exponentiation, including ordinal versions of tetration, pentation, and hexation. See also Veblen function.

As discussed above, the Cantor normal form of ordinals below ε0 can be expressed in an alphabet containing only the function symbols for addition, multiplication and exponentiation, as well as constant symbols for each natural number and for ω. We can do away with the infinitely many numerals by using just the constant symbol 0 and the operation of successor, S (for example, the natural number 4 may be expressed as S(S(S(S(0))))). This describes an ordinal notation: a system for naming ordinals over a finite alphabet. This particular system of ordinal notation is called the collection of arithmetical ordinal expressions, and can express all ordinals below ε0, but cannot express ε0. There are other ordinal notations capable of capturing ordinals well past ε0, but because there are only countably many finite length strings over any finite alphabet, for any given ordinal notation there will be ordinals below ω1 (the first uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions, and more general primitive recursive ordinal functions can be used to describe larger ordinals.

There are arithmetic operations on ordinals by virtue of the one to one correspondence between ordinals and nimbers. Three common operations on nimbers are nimber addition, nimber multiplication, and minimum excludance (mex). Nimber addition is a generalization of the bitwise exclusive or operation on natural numbers. The mex of a set of ordinals is the smallest ordinal not present in the set.