\begin{definition}[Definition:Ascending Chain Condition]
Let $\struct {P, \le}$ be an ordered set.
Then $S$ is said to have the '''ascending chain condition (ACC)''' {{iff}} every increasing sequence $x_1 \le x_2 \le x_3 \le \cdots$ with $x_i \in P$ eventually terminates: there is $n \in \N$ such that $x_n = x_{n + 1} = \cdots$.
\end{definition}