Lévy reflection theorem

For every finite collection $\Phi$ of first-order formulas, there are arbitrarily large ordinals $\alpha$ such that, for every $\phi \in \Phi$ and all parameters in $V_{\alpha }$,The reflecting ordinals for $\Phi$ form a closed unbounded class.

For inaccessible $\kappa$ and any predicate $R\subseteq V_{\kappa }$, the ordinals $\alpha <\kappa$ with $⟨V_{\alpha },\in ,R\cap V_{\alpha }⟩\prec ⟨V_{\kappa },\in ,R⟩$ form a club. Closure under ranks of existential witnesses gives unboundedness, and the elementary chain theorem gives closedness.

A finite collection of first-order formulas has a closed unbounded class of ordinals whose levels in a definable continuous hierarchy agree with its union for all parameters in the level. Close the collection under subformulas, bound the least witness levels for existential instances on each set-sized stage by Axiom schema of replacement, and iterate these bounds countably. Continuity supplies the closed stages, and induction on formulas proves agreement. The Tarski-Vaught test describes the same witness criterion. For a hierarchy of length an uncountable regular cardinal, bounds stay below that cardinal when every stage has smaller cardinality.

A first-order formula $\phi$ describes an ordinal $\alpha$ when $\alpha$ is the least ordinal such that $V_{\alpha }\models \phi$. No strongly inaccessible cardinal can be described: if $V_{\kappa }\models \phi$, the Lévy reflection theorem produces some $\alpha <\kappa$ with $V_{\alpha }\models \phi$.