Prove that if $G:$ On $\times V \rightarrow V$ is a definable function, then there is a definable function $F:$ On $\rightarrow V$ satisfying

$$F(\alpha)=G(\alpha,\{F(\beta): \beta<\alpha\})$$

Define the notion of an initial ordinal, and explain its significance for cardinal arithmetic. State Hartogs' lemma. Using the recursion theorem define, giving justification, a function $\omega:$ On $\rightarrow$ On which enumerates the infinite initial ordinals.

Is there an ordinal $\alpha$ with $\alpha=\omega_{\alpha} ?$ Justify your answer.