I was looking at Woodin's chapter in the book Infinity: New Research Frontiers and I'm confused by some of his remarks regarding class-sized elementary embeddings. The following is taken from the book:
I have two questions:
Why is there a bound on the complexity of the defining formulas?
I don't see why the two bulleted items are equivalent. For the forward direction, fix an ordinal $\alpha$ and $a\in V_\alpha$, then by the absoluteness of satisfaction and because $\phi$ is $\Delta_0$-definable,
$$ V_\alpha\models \phi[a]\iff V\models (V_\alpha\models \phi[a]) \iff M\models (j(V_\alpha)\models \phi [j(a)]) \\ \iff j(V_\alpha)\models \phi[j(a)] $$
I get stuck on the backwards direction. I'm sure it's some kind of reflection argument, but the details escape me. Given a formula $\phi$ and $a\in V$, let $\alpha>rk(a)$ reflect $\phi$, i.e. $V_\alpha \preceq_\phi V$. Then $$ V\models \phi[a] \iff V_\alpha\models \phi[a] \iff j(V_\alpha)\models \phi[j(a)] $$
If $M\models ZFC$, then I can argue that $M\models V_\alpha\preceq_\phi V$ so that $V_\alpha^M\preceq_\phi M$, and so it suffices to show that $V_\alpha^M=j(V_\alpha)$ for every ordinal $\alpha$, but I'm not sure how to do that (another reflection argument?)