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| In [[set theory]], a '''Rowbottom cardinal''', introduced by {{harvs|txt|authorlink=Frederick Rowbottom|last=Rowbottom|year=1971}}, is a certain kind of [[large cardinal]] number.
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| An [[uncountable set|uncountable]] [[cardinal number]] κ is said to be '''''Rowbottom''''' if for every function ''f'': [κ]<sup><ω</sup> → λ (where λ < κ) there is a set ''H'' of order type κ that is quasi-[[Homogeneous (large cardinal property)|homogeneous]] for ''f'', i.e., for every ''n'', the ''f''-image of the set of ''n''-element subsets of ''H'' has [[countable|countably]] many elements.
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| Every [[Ramsey cardinal]] is Rowbottom, and every Rowbottom cardinal is [[Jónsson cardinal|Jónsson]]. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent.
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| In general, Rowbottom cardinals need not be [[large cardinal]]s in the usual sense: Rowbottom cardinals could be [[singular cardinal|singular]]. It is an open question whether ZFC + “<math>\aleph_{\omega}</math> is Rowbottom” is consistent. If it is, it has much higher consistency strength than the existence of a Rowbottom cardinal. The [[axiom of determinacy]] does imply that <math>\aleph_{\omega}</math> is Rowbottom (but contradicts the [[axiom of choice]]).
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| == References ==
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| * {{cite book|last=Kanamori|first=Akihiro|year=2003|publisher=Springer|title=The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings|edition=2nd ed|isbn=3-540-00384-3}}
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| *{{Citation | last1=Rowbottom | first1=Frederick | title=Some strong axioms of infinity incompatible with the axiom of constructibility | origyear=1964 | doi=10.1016/0003-4843(71)90009-X | id={{MathSciNet | id = 0323572}} | year=1971 | journal=Annals of Pure and Applied Logic | issn=0168-0072 | volume=3 | issue=1 | pages=1–44}}
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| [[Category:Large cardinals]]
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| {{settheory-stub}}
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