# Definition A [[forcing#^e01f21|forcing notion]] $\mathbb{P}$ satisfies *Axiom $A$* if there is a collection $\left\{\leq_n\right\}_{n=0}^{\infty}$ of partial orderings of $\mathbb{P}$ such that $p \leq_0 q$ implies $p \leq q$ and for every $n, p \leq_{n+1} q$ implies $p \leq_n q$, and 1. if $\left\langle p_n: n \in \omega\right\rangle$ is a sequence such that $p_0 \geq_0 p_1 \geq_1 \ldots \geq_{n-1} p_n \geq_n \ldots$ then there is a $q$ such that $q \leq_n p_n$ for all $n$; 2. for every $p \in P$, for every $n$ and for every ordinal name $\dot{\alpha}$ there exist a $q \leq_n p$ and a countable set $B$ such that $q \Vdash \dot{\alpha} \in B$. # Properties # Relations ## Implied by - [[Chain conditions|c.c.c]] - [[Closed forcing|Countably closed]] ## Implies - [[Proper forcing]]