Let $\gamma$ be an initial infinite ordinal. Show that $\gamma=\omega_{\alpha}$ for some ordinal $\alpha$
The definition I use for $\omega_{\alpha}$ as follows: Suppose $\omega_{\beta},\aleph_{\beta}$ have been defined for all $\beta<\alpha$. By transfinite induction, we define $$\omega_{\alpha}=\{\gamma \ \text{is ordinal} : |\gamma|\le\aleph_{\beta} \ \text{for some} \ \beta<\alpha\}$$