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Ordinal 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 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.
The sum of two well-ordered sets S and T is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products S × {0} and T × {1}. This way, every element of S is smaller than every element of T, comparisons within S keep the order they already have, and likewise for comparisons within T.
The definition of ordinary addition α + β can also be given by transfinite recursion on β. When the right addend β = 0, addition gives α + 0 = α for any α. For β > 0, the value of α + β is the smallest ordinal strictly greater than the sum of α and δ for all δ < β. Writing the successor and limit ordinals cases separately:
Ordinal addition on the natural numbers is the same as standard addition. The first transfinite ordinal is ω, the set of all natural numbers, followed by ω + 1, ω + 2, etc. The ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0′ < 1′ < 2′ < ... for the second copy, ω + ω looks like
This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0′ do not have direct predecessors.
Ordinal addition is, in general, not commutative. For example, 3 + ω = ω since the order relation for 3 + ω is 0 < 1 < 2 < 0′ < 1′ < 2′ < ..., which can be relabeled to ω. In contrast ω + 3 is not equal to ω since the order relation 0 < 1 < 2 < ... < 0′ < 1′ < 2′ has a largest element (namely, 2′) and ω does not (ω and ω + 3 are equipotent, but not order-isomorphic).
Ordinal addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.
Addition is strictly increasing and continuous in the right argument:
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Ordinal 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 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.
The sum of two well-ordered sets S and T is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products S × {0} and T × {1}. This way, every element of S is smaller than every element of T, comparisons within S keep the order they already have, and likewise for comparisons within T.
The definition of ordinary addition α + β can also be given by transfinite recursion on β. When the right addend β = 0, addition gives α + 0 = α for any α. For β > 0, the value of α + β is the smallest ordinal strictly greater than the sum of α and δ for all δ < β. Writing the successor and limit ordinals cases separately:
Ordinal addition on the natural numbers is the same as standard addition. The first transfinite ordinal is ω, the set of all natural numbers, followed by ω + 1, ω + 2, etc. The ordinal ω + ω is obtained by two copies of the natural numbers ordered in the usual fashion and the second copy completely to the right of the first. Writing 0′ < 1′ < 2′ < ... for the second copy, ω + ω looks like
This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0′ do not have direct predecessors.
Ordinal addition is, in general, not commutative. For example, 3 + ω = ω since the order relation for 3 + ω is 0 < 1 < 2 < 0′ < 1′ < 2′ < ..., which can be relabeled to ω. In contrast ω + 3 is not equal to ω since the order relation 0 < 1 < 2 < ... < 0′ < 1′ < 2′ has a largest element (namely, 2′) and ω does not (ω and ω + 3 are equipotent, but not order-isomorphic).
Ordinal addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.
Addition is strictly increasing and continuous in the right argument: