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| In [[mathematics]], specifically in [[axiomatic set theory]], a '''Hartogs number''' is a particular kind of [[cardinal number]]. It was shown by [[Friedrich Hartogs]] in 1915, from [[Zermelo-Fraenkel set theory|ZF]] alone (that is, without using the [[axiom of choice]]), that there is a least [[well-ordered]] [[cardinal number|cardinal]] greater than a given well-ordered cardinal.
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| To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: If ''X'' is any set, then the Hartogs number of ''X'' is the least [[ordinal number|ordinal]] α such that there is no [[Injective function|injection]] from α into ''X''. If ''X'' cannot be well-ordered, then we can no longer say that this α is the least well-ordered cardinal ''greater'' than the cardinality of ''X'', but it remains the least well-ordered cardinal ''not less than or equal to'' the cardinality of ''X''. The [[map (mathematics)|map]] taking ''X'' to α is sometimes called '''Hartogs' function'''.
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| ==Proof==
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| Given some basic theorems of set theory, the proof is simple. Let <math>\alpha = \{\beta \in \textrm{Ord}| \exists i: \beta \hookrightarrow X\}</math>. First, we verify that α is a set.
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| #''X'' × ''X'' is a set, as can be seen in [[axiom of power set#Consequences|axiom of power set]].
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| # The [[power set]] of ''X'' × ''X'' is a set, by the [[axiom of power set]].
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| # The class ''W'' of all [[reflexive relation|reflexive]] well-orderings of subsets of ''X'' is a definable subclass of the preceding set, so it is a set by the [[axiom schema of separation]].
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| # The class of all [[order type]]s of well-orderings in ''W'' is a set by the [[axiom schema of replacement]], as
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| #::([[Domain (mathematics)|Domain]](''w''), ''w'') <math>\cong</math> (β, ≤)
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| #:can be described by a simple formula.
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| But this last set is exactly α.
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| Now because a [[transitive set]] of ordinals is again an ordinal, α is an ordinal. Furthermore, if there were an injection from α into ''X'', then we would get the contradiction that α ∈ α. It is claimed that α is the least such ordinal with no injection into ''X''. Given β < α, β ∈ α so there is an injection from β into ''X''.
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| ==References==
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| *{{Cite journal
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| | last = Hartogs
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| | first = Fritz
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| | author-link =
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| | title = Über das Problem der Wohlordnung
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| | journal = [[Mathematische Annalen]]
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| | language = [[German language|German]]
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| | volume = 76
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| | pages =438–443
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| | year = 1915
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| | url = http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002266105
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| | doi = 10.1007/BF01458215
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| | id =
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| | jfm = 45.0125.01
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| | issue = 4
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
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| }}. Available at the [http://www.digizeitschriften.de/ DigiZeitschriften].
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| * {{cite book|authorlink=Thomas Jech|author=Jech, Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2}}
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| * {{cite web | title=Axiomatic set theory | work=Course Notes | author=Charles Morgan | publisher=University of Bristol | url=http://www.ucl.ac.uk/~ucahcjm/ast/ast_notes_4.pdf | accessdate =2010-04-10 }}
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| [[Category:Set theory]]
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| [[Category:Cardinal numbers]]
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| {{settheory-stub}}
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