# Definitions ## The full tower The *(full) stationary tower* on an ordinal $\delta$, denoted $\mathbb{P}_{<\delta}$ is the [[forcing|poset]] with: - Conditions: $a\in V_{\delta}$ which is a [[Club sets and stationary sets|stationary]] subset of $\mathcal{P}(\cup a)$ - Order: $a\geq b$ iff $\cup a \subseteq \cup b$ and $\forall Z\in b$ $Z\cap\left(\cup a\right)\in a$ (i.e. $b\downarrow\left(\cup a\right)\subseteq a$). ## The countable tower The *countable stationary tower* on an ordinal $\delta$, denoted $\mathbb{Q}_{<\delta}$ is the [[forcing|poset]] with: - Conditions: $a\in V_{\delta}$ which is a [[Club sets and stationary sets|stationary]] subset of $\mathcal{P}_{\omega_{1}}(\cup a)$ - Order: $a\geq b$ iff $\cup a \subseteq \cup b$ and $\forall Z\in b$ $Z\cap\left(\cup a\right)\in a$ (i.e. $b\downarrow\left(\cup a\right)\subseteq a$). # Properties ## The full tower