Let $B$ be an infinite, complete Boolean algebra, and let $\kappa = \operatorname{sat}(B)$. I would like to show that $\kappa$ is uncountable. If we suppose $\kappa$ is countable, that is to say $\kappa = \aleph_0$, then there must exist no descending countable chains in $B$. This means that we can only construct finite descending chains $u_0 > \ldots > u_n$ in $B$.
I think that this tells us $B$ must be finite, but I can't quite see why. Is it to do with the fact that we can construct a partition $W_n$ of $B$ for each $0