In set theory, a Rowbottom cardinal, introduced by Frederick Rowbottom, is a certain kind of large cardinal number. An uncountable cardinal number κ {\displaystyle \kappa } is said to be λ-Rowbottom if for every function f : [ κ ] < ω → λ {\displaystyle f:[\kappa ]^{<\omega }\to \lambda } (where λ < κ {\displaystyle \lambda <\kappa } ) there is a set H {\displaystyle H} of order type κ {\displaystyle \kappa } that is quasi-homogeneous for f {\displaystyle f} , i.e., for every n {\displaystyle n} , the f {\displaystyle f} -image of the set of n {\displaystyle n} -element subsets of H {\displaystyle H} has < λ {\displaystyle <\lambda } elements. κ {\displaystyle \kappa } is simply Rowbottom if it is ω1-Rowbottom. Every Ramsey cardinal is Rowbottom, and every Rowbottom cardinal is Jónsson. By a theorem of Kleinberg, the theories ZFC + “there is a Rowbottom cardinal” and ZFC + “there is a Jónsson cardinal” are equiconsistent. In general, Rowbottom cardinals need not be large cardinals in the usual sense: Rowbottom cardinals could be singular. It is an open question whether ZFC + “ ℵ ω {\displaystyle \aleph _{\omega }} 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 ℵ ω {\displaystyle \aleph _{\omega }} is Rowbottom (but contradicts the axiom of choice).
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