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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 which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called "natural" arithmetical operations retain commutativity at the expense of continuity. == Addition == The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal which results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordered set ''T'', meaning one defines an order on ''S'' ''T'' in which every element of ''S'' is smaller than every element of ''T''. The sets ''S'' and ''T'' themselves keep the ordering they already have. This addition of the order-types is associative and generalizes the addition of natural numbers. The first transfinite ordinal is ω, the set of all natural numbers. For example, 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 :0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ... 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. As another example, here are 3 + ω and ω + 3: :0 < 1 < 2 < 0' < 1' < 2' < ... :0 < 1 < 2 < ... < 0' < 1' < 2' After relabeling, the former just looks like ω itself, i.e. 3 + ω = ω, while the latter does not: is not equal to ω since has a largest element (namely, 2') and ω does not. Hence, this addition is not commutative. In fact it is quite rare for α+β to be equal to β+α: this happens if and only if α=γ''m'', β=γ''n'' for some ordinal γ and natural numbers ''m'' and ''n''. Moreover the relation α+β = β+α is an equivalence relation on the set of nonzero ordinals, and all the equivalence classes are countable infinite. However, addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω. The definition of addition can also be given inductively (the following induction is on ''β''): * ''α'' + 0 = ''α'', * ''α'' + (''β'' + 1) = (''α'' + ''β'') + 1 (here, "+ 1" denotes the ''successor'' of an ordinal), * and if ''β'' is a limit ordinal then ''α'' + ''β'' is the limit of the ''α'' + ''δ'' for all ''δ'' < ''β''. Using this definition, ω + 3 can be seen to be a successor ordinal (it is the successor of ω + 2), whereas 3 + ω is a limit ordinal, namely, the limit of 3 + 0 = 3, 3 + 1 = 4, 3 + 2 = 5, etc., which is just ω. Zero is an additive identity ''α'' + 0 = 0 + ''α'' = ''α''. Addition is associative (''α'' + ''β'') + ''γ'' = ''α'' + (''β'' + ''γ''). Addition is strictly increasing and continuous in the right argument: : but the analogous relation does not hold for the left argument; instead we only have: : Ordinal addition is left-cancellative: if ''α'' + ''β'' = ''α'' + ''γ'', then ''β'' = ''γ''. Furthermore, one can define left subtraction for ordinals ''β'' ≤ ''α'': there is a unique ''γ'' such that ''α'' = ''β'' + ''γ''. On the other hand, right cancellation does not work: : but Nor does right subtraction, even when ''β'' ≤ ''α'': for example, there does not exist any ''γ'' such that ''γ'' + 42 = ω. If the ordinals less than α are closed under addition and contain 0 then α is occasionally called a γ-number (see additively indecomposable ordinal). These are exactly the ordinals of the form ωβ. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「ordinal arithmetic」の詳細全文を読む スポンサード リンク
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