Rowbottom cardinal

• Definition
• Facts

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)

Definition

Rowbottom cardinals are defined with a partition property:

• $\kappa$ is $\nu$-Rowbottom iff $\kappa\rightarrow [\kappa]^{<\omega}_{\lambda,<\nu}$ for every $\lambda<\kappa$. This means that for any partition (function) $f:[\kappa]^{<\omega}\rightarrow\lambda$, there is some set of ordinals $H\subseteq\kappa$ such that $(H,<)$ has order type $\kappa$ and $|f”[H]^{<\omega}|<\nu$.
• $\kappa$ is Rowbottom iff it is $\omega_1$-Rowbottom.

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)

Facts

• If a Rowbottom exists, then $0^{#}$ exists and is consistent. (Kanamori, 2009)
• Every Rowbottom cardinal is Jónsson. (Kanamori, 2009)
• Every Rowbottom cardinal $\kappa$ either has cofinality $\omega$ or is weakly inaccessible. (Kanamori, 2009)
• Every $\nu$-Rowbottom cardinal either has cofinality less than $\nu$ or is weakly inaccessible (and thus if a $\nu$-Rowbottom cardinal $\kappa$ has cofinality $\nu$, then $\nu=\kappa$ and $\kappa$ is $\kappa$-Rowbottom.) (Kanamori, 2009)
• Any singular limit $\kappa$ of measurable cardinals is $\mathrm{cf}(\kappa)^+$-Rowbottom. (Kanamori, 2009)
• If $\kappa=2^{<\nu}$ is a regular $\nu$-Rowbottom cardinal, then for any $\nu\leq\lambda<\kappa$, $2^\lambda=\kappa$. Thus, if the first condition holds for $\kappa$ and $\nu$ but $\nu < \kappa$, then GCH fails at every cardinal $\lambda\in[\nu,\kappa)$. (Kanamori, 2009)
• If $\kappa$ is $\nu$-Rowbottom and there is a limit cardinal $\lambda$ such that $\nu\leq\lambda<\kappa$, then $\kappa$ is a limit of limit cardinals (i.e. $\aleph_{\alpha^\beta}$ for some ordinals $\alpha$ and $\beta$). (Kanamori, 2009)

References

1. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
2. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf

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