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| In [[mathematics]], '''limit cardinals''' are certain [[cardinal number]]s. A cardinal number λ is a '''weak limit cardinal''' if λ is neither a [[successor cardinal]] nor zero. This means that one cannot "reach" λ by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
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| A cardinal λ is a '''strong limit cardinal''' if λ cannot be reached by repeated [[Power set|powerset]] operations. This means that λ is nonzero and, for all κ < λ, 2<sup>κ</sup> < λ. Every strong limit cardinal is also a weak limit cardinal, because κ<sup>+</sup> ≤ 2<sup>κ</sup> for every cardinal κ, where κ<sup>+</sup> denotes the successor cardinal of κ.
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| The first infinite cardinal, <math>\aleph_0</math> ([[Aleph number#Aleph-naught|aleph-naught]]), is a strong limit cardinal, and hence also a weak limit cardinal.
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| == Constructions ==
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| One way to construct limit cardinals is via the union operation: <math>\aleph_{\omega}</math> is a weak limit cardinal, defined as the union of all the alephs before it; and in general <math>\aleph_{\lambda}</math> for any [[limit ordinal]] λ is a weak limit cardinal.
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| The [[beth number|ב operation]] can be used to obtain strong limit cardinals. This operation is a map from ordinals to cardinals defined as
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| :<math>\beth_{0} = \aleph_0,</math>
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| :<math>\beth_{\alpha+1} = 2^{\beth_{\alpha}},</math> (the smallest ordinal [[equinumerous]] with the powerset)
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| :If λ is a limit ordinal, <math>\beth_{\lambda} = \bigcup \{ \beth_{\alpha} : \alpha < \lambda\}.</math>
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| The cardinal
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| :<math>\beth_{\omega} = \bigcup \{ \beth_{0}, \beth_{1}, \beth_{2}, \ldots \} = \bigcup_{n < \omega} \beth_{n} </math>
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| is a strong limit cardinal of [[cofinality]] ω. More generally, given any ordinal α, the cardinal
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| :<math>\beth_{\alpha+\omega} = \bigcup_{n < \omega} \beth_{\alpha+n} </math>
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| is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.
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| == Relationship with ordinal subscripts ==
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| If the [[axiom of choice]] holds, every cardinal number has an [[initial ordinal]]. If that initial ordinal is <math>\omega_{\lambda} \,,</math> then the cardinal number is of the form <math>\aleph_\lambda</math> for the same ordinal subscript λ. The ordinal λ determines whether <math>\aleph_\lambda</math> is a weak limit cardinal. Because <math>\aleph_{\alpha^+} = (\aleph_\alpha)^+ \,,</math> if λ is a successor ordinal then <math>\aleph_\lambda</math> is not a weak limit. Conversely, if a cardinal κ is a successor cardinal, say <math>\kappa = (\aleph_{\alpha})^+ \,,</math> then <math>\kappa = \aleph_{\alpha^+} \,.</math> Thus, in general, <math>\aleph_\lambda</math> is a weak limit cardinal if and only if λ is zero or a limit ordinal.
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| Although the ordinal subscript tells whether a cardinal is a weak limit, it does not tell whether a cardinal is a strong limit. For example, [[Zermelo–Fraenkel set theory|ZFC]] proves that <math>\aleph_\omega</math> is a weak limit cardinal, but neither proves nor disproves that <math>\aleph_\omega</math> is a strong limit cardinal (Hrbacek and Jech 1999:168). The [[generalized continuum hypothesis]] states that <math>\kappa^+ = 2^{\kappa} \,</math> for every infinite cardinal κ. Under this hypothesis, the notions of weak and strong limit cardinals coincide.
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| == The notion of inaccessibility and large cardinals ==
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| The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using [[cofinality]]. For a weak (resp. strong) limit cardinal κ the requirement that cf(κ) = κ (i.e. κ be [[regular cardinal|regular]]) so that κ cannot be expressed as a sum (union) of fewer than κ smaller cardinals. Such a cardinal is called a [[inaccessible cardinal|weakly (resp. strongly) inaccessible cardinal]]. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.
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| <math>\aleph_0</math> would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above <math>\aleph_0</math>, due to [[Gödel's incompleteness theorems|Gödel's Incompleteness Theorem]]. More specifically, if <math>\kappa</math> is weakly inaccessible then <math>L_{\kappa} \models ZFC</math>. These form the first in a hierarchy of [[large cardinal]]s.
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| == See also ==
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| * [[Cardinal number]]
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| == References ==
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| * {{citation
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| | last1= Hrbacek | first1 = Karel
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| | last2=Jech | first2=Thomas
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| | title = Introduction to Set Theory
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| | edition = 3
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| | year = 1999 | isbn = 0-8247-7915-0
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| }}
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| * {{Citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=third millennium | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | doi=10.1007/3-540-44761-X | year=2003}}
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| * {{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | title=Set theory: An introduction to independence proofs | publisher=[[Elsevier]] | isbn=978-0-444-86839-8 | year=1980}}
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| ==External links==
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| * http://www.ii.com/math/cardinals/ Infinite ink on cardinals
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| [[Category:Set theory]]
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| [[Category:Cardinal numbers]]
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