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Limit ordinal
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 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, a successor ordinal, or a limit ordinal.
For example, the smallest limit ordinal is ω, the smallest ordinal greater than every natural number. This 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. The next-smallest limit ordinal is ω+ω. This will be discussed further in the article.
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.
Various other ways to define limit ordinals are:
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 while others exclude it.
Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n. Taking the union (the supremum operation on any set of ordinals) of all the ω·n, we get ω·ω = ω2, which generalizes to ωn for any natural number n. This process can be further iterated as follows to produce:
In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still countable ordinals. However, there is no recursively enumerable scheme for systematically naming all ordinals less than the Church–Kleene ordinal, which is a countable ordinal.
Beyond the countable, the first uncountable ordinal is usually denoted ω1. It is also a limit ordinal.
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Limit ordinal
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 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, a successor ordinal, or a limit ordinal.
For example, the smallest limit ordinal is ω, the smallest ordinal greater than every natural number. This 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. The next-smallest limit ordinal is ω+ω. This will be discussed further in the article.
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.
Various other ways to define limit ordinals are:
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 while others exclude it.
Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·n for any natural number n. Taking the union (the supremum operation on any set of ordinals) of all the ω·n, we get ω·ω = ω2, which generalizes to ωn for any natural number n. This process can be further iterated as follows to produce:
In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still countable ordinals. However, there is no recursively enumerable scheme for systematically naming all ordinals less than the Church–Kleene ordinal, which is a countable ordinal.
Beyond the countable, the first uncountable ordinal is usually denoted ω1. It is also a limit ordinal.