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| In the theory of [[cardinal numbers]], we can define a '''successor''' operation similar to that in the [[ordinal number]]s. This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same [[cardinality]] (a [[bijection]] can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above; in the style of Hilbert's [[Hilbert's paradox of the Grand Hotel|Hotel Infinity]]). Using the [[von Neumann cardinal assignment]] and the [[axiom of choice]] (AC), this successor operation is easy to define: for a cardinal number κ we have
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| :<math>\kappa^+ = |\inf \{ \lambda \in ON \ |\ \kappa < |\lambda| \}|</math> ,
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| where ON is the class of ordinals. That is, the successor cardinal is the cardinality of the least ordinal into which a set of the given cardinality can be mapped one-to-one, but which cannot be mapped one-to-one back into that set.
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| That the set above is nonempty follows from [[Hartogs number|Hartogs' theorem]], which says that for any [[well-order]]able cardinal, a larger such cardinal is constructible. The minimum actually exists because the ordinals are well-ordered. It is therefore immediate that there is no cardinal number in between κ and κ<sup>+</sup>. A '''successor cardinal''' is a cardinal which is κ<sup>+</sup> for some cardinal κ. In the infinite case, the successor operation skips over many ordinal numbers; in fact, every infinite cardinal is a [[limit ordinal]]. Therefore, the successor operation on cardinals gains a lot of power in the infinite case (relative the ordinal successorship operation), and consequently the cardinal numbers are a very "sparse" subclass of the ordinals. We define the sequence of [[aleph number|alephs]] (via the [[axiom of replacement]]) via this operation, through all the ordinal numbers as follows:
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| :<math>\aleph_0 = \omega</math>
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| :<math>\aleph_{\alpha+1} = \aleph_{\alpha}^+</math>
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| and for λ an infinite limit ordinal,
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| :<math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta</math> | |
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| If β is a [[successor ordinal]], then <math>\aleph_{\beta}</math> is a successor cardinal. Cardinals which are not successor cardinals are called [[limit cardinal]]s; and by the above definition, if λ is a limit ordinal, then <math>\aleph_{\lambda}</math> is a limit cardinal.
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| The standard definition above is restricted to the case when the cardinal can be well-ordered, i.e. is finite or an aleph. Without the axiom of choice, there are cardinals which cannot be well-ordered. Some mathematicians have defined the successor of such a cardinal as the cardinality of the least ordinal which cannot be mapped one-to-one into a set of the given cardinality. That is:
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| :<math>\kappa^+ = |\inf \{ \lambda \in ON \ |\ |\lambda| \nleq \kappa \}|</math> .
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| ==See also==
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| *[[Cardinal assignment]]
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| == References ==
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| *[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
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| *[[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
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| *[[Kenneth Kunen|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.
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| [[Category:Set theory]]
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| [[Category:Cardinal numbers]]
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