Rowbottom cardinals were discovered by Frederick Rowbottom in 1971 as a strong [large cardinal axiom](Upper_attic.md) which implies the existence and consistency of $0^{\sharp}$ ([[Constructible universe#Sharps|zero-sharp]]).
In terms of consistency strength, [ZFC](ZFC.md "ZFC") + Rowbottom is equiconsistent to ZFC + [Jónsson](Jonsson.md "Jonsson"), ZFC + Rowbottom is equiconsistent to ZFC + [Ramsey](Ramsey.md "Ramsey"),
and ZFC + Rowbottom is stronger than ZFC + $0^{\#}$. Every Rowbottom cardinal is Jónsson, and every Ramsey cardinal is Rowbottom. {% cite Kanamori2009 %}
## Definition
Rowbottom cardinals are defined with a [](Partition%20property.md):
- $\kappa$ 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.md),
namely, any model of type $(\kappa,\lambda)$ for some uncountable
$\lambda<\kappa$ has a proper elementary substructure of type
$(\kappa,\aleph_0)$. {% cite Jech2003 %}
Rowbottom cardinals are not necessarily "large". In fact, the
<a href="Axiom_of_Determinacy" class="mw-redirect" title="Axiom of Determinacy">Axiom of Determinacy</a>
implies $\aleph_\omega$ is Rowbottom, and it is widely considered
consistent for $\aleph_\omega$ to be Rowbottom even under the
<a href="Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a>.
If it is consistent for $\aleph_\omega$ to be Rowbottom, it is
consistent for $\aleph_{\omega^2}$ to be the least Rowbottom
cardinal. {% cite Kanamori2009 %}
## Facts
- If a Rowbottom exists, then
<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^{\#}
lt;/a>
exists and is consistent.
{% cite Kanamori2009 %}
- Every Rowbottom cardinal is Jónsson.
{% cite Kanamori2009 %}
- Every Rowbottom cardinal $\kappa$ either has cofinality $\omega$
or is weakly inaccessible.
{% cite Kanamori2009 %}
- Every $\nu$-Rowbottom cardinal either has cofinality less than
$\nu$ or is [](Inaccessible.md)
(and thus if a $\nu$-Rowbottom cardinal $\kappa$ has cofinality
$\nu$, then $\nu=\kappa$ and $\kappa$ is $\kappa$-Rowbottom.)
{% cite Kanamori2009 %}
- Any singular limit $\kappa$ of
[measurable](Measurable.md "Measurable")
cardinals is $\mathrm{cf}(\kappa)^+$-Rowbottom.
{% cite Kanamori2009 %}
- 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
<a href="GCH" class="mw-redirect" title="GCH">GCH</a>
fails at every cardinal $\lambda\in\[\nu,\kappa)$.
{% cite Kanamori2009 %}
- 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 $\beta$).
{% cite Kanamori2009 %}