Hi. Can you explain why $$\DeclareMathOperator{cf}{cf}\cf(\aleph_\alpha)=\begin{cases}\aleph_\alpha & \alpha=\beta+1\\ \cf(\alpha) & \text{otherwise}\end{cases}$$ I can see why it is true in the successor case but not the limit case. I get that $\cf(\aleph_\alpha)\le\cf(\alpha)$ but why can't it be strictly smaller? — Elie Bergman Jul 4, 2016 at 15:54