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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 and only if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal. For example, ω, the smallest ordinal greater than every natural number is a limit ordinal because for any smaller ordinal (i.e., for any natural number) ''n'' we can find another natural number larger than it (e.g. ''n''+1), but still less than ω. Using the Von Neumann definition of ordinals, every ordinal is the well-ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal. ==Alternative definitions== Various other ways to define limit ordinal are: *It is equal to the supremum of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.) *It is not zero and has no maximum element. *It can be written in the form ωα for α > 0. That is, in the Cantor normal form there is no finite number as last term, and the ordinal is nonzero. *It is a limit point of the class of ordinal numbers, with respect to the order topology. (The other ordinals are isolated points.) Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor; some textbooks include 0 in the class of limit ordinals〔for example, Thomas Jech, ''Set Theory''. Third Millennium edition. Springer.〕 while others exclude it.〔for example, Kenneth Kunen, ''Set Theory. An introduction to independence proofs''. North-Holland.〕 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「Limit ordinal」の詳細全文を読む スポンサード リンク
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