Rowbottom cardinal

From Wikipedia, the free encyclopedia

In set theory, a Rowbottom cardinal (named after Frederick Rowbottom) is a certain kind of large cardinal number.

An uncountable cardinal number κ is said to be Rowbottom if for every function f: [κ] ^{< ω} → λ (where λ < κ) there is a set H of order type κ that is quasi-homogeneous for f, i.e., for every n, the f-image of the set of n-element subsets of H has countably many elements.

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+“ $\aleph_{\omega}$ 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 $\aleph_{\omega}$ is Rowbottom (but contradicts the axiom of choice).