## Franklin's Notes

### Limits of ordinals

Consider a transfinite ordinal-indexed sequence of ordinal numbers for $\xi<\varphi$ for some transfinite ordinal $\varphi$. Let $\lambda$ be the smallest ordinal which is greater than all of the $\alpha_\xi$ (some such ordinal certainly exists, since the successor of the union of the $\alpha_\xi$ is one example). If $\varphi$ is a limit ordinal, and for any $\mu<\lambda$, there exists $\nu <\varphi$ such that $\mu <\alpha_\xi<\lambda$ for all $\nu\leq\xi < \varphi$, then we say that $\lambda$ is the limit of the sequence $(\alpha_\xi)_{\xi<\varphi}$, and write