Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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Rowbottom cardinal

Rowbottom cardinals were discovered by Frederick Rowbottom in 1971 as a strong large cardinal axiom which implies the existence and consistency of $0^{#}$. In terms of consistency strength, ZFC + Rowbottom is equiconsistent to ZFC + Jónsson, ZFC + Rowbottom is equiconsistent to ZFC + Ramsey, and ZFC + Rowbottom is stronger than ZFC + $0^{\#}$. Every Rowbottom cardinal is Jónsson, and every Ramsey cardinal is Rowbottom. (Kanamori, 2009)


Rowbottom cardinals are defined with a partition property:

Equivalently, $\kappa$ is Rowbottom if and only if $\kappa>\aleph_1$ and $\kappa$ satisfies a strong generalization of Chang’s conjecture, namely, any model of type $(\kappa,\lambda)$ for some uncountable $\lambda<\kappa$ has a proper elementary substructure of type $(\kappa,\aleph_0)$. (Jech, 2003)

Rowbottom cardinals are not necessarily “large”. In fact, the Axiom of Determinacy implies $\aleph_\omega$ is Rowbottom, and it is widely considered consistent for $\aleph_\omega$ to be Rowbottom even under the Axiom of Choice. If it is consistent for $\aleph_\omega$ to be Rowbottom, it is consistent for $\aleph_{\omega^2}$ to be the least Rowbottom cardinal. (Kanamori, 2009)



  1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  2. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag.
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