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. [1]
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 $|f^{\prime\,\prime}[H]^{<\omega}|<\nu$.
- $\kappa$ 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 $(\kappa,\aleph_0)$. [2]
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. [1]
Facts
- Every Rowbottom cardinal $\kappa$ either has cofinality $\omega$ or is weakly inaccessible. [1]
- 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.) [1]
- Any singular limit $\kappa$ of measurable cardinals is $\mathrm{cf}(\kappa)^+$-Rowbottom. [1]
- 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 $\beta$). [1]
Consistency strength
- If a Rowbottom exists, then $0^{\#}$ exists and is consistent. [1]
- Every Rowbottom cardinal is Jónsson. [1]
- 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 $\lambda\in[\nu,\kappa)$. [1]
- For any $0<n<\omega$, the theories $\textrm{ZF+DC}_{\aleph_n}+\aleph_\omega\textrm{ is Rowbottom and carries a Rowbottom filter}$ and $\textrm{ZFC+A measurable exists}$ are equiconsistent. [1]
References
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex