# What is the consistency strength of adding this ordinal reflection scheme on top of Ackermann set theory?

Axiom scheme of Ordinal Reflection: if $$\phi$$ is a formula that doesn't use the symbol $$V$$, whose parameters are among $$x_1,..,x_n$$; then: $$\forall x_1 \in V,\dotsc,\forall x_n \in V: \phi(On) \to \\\forall c \subseteq On \\ (c=\{\alpha \in On| \phi(\alpha)\} \lor \bigcup( c) = On \to |c|=On);$$ is an axiom.

Where: $$On=\{\alpha \in V|\operatorname{ordinal}(\alpha)\};$$ and "$$|c|$$" stands for cardinality of class $$c$$, defined as the least von Neumann ordinal class bijective to $$c$$.

Add this axiom on top of Ackermann's set theory.

What would be the consistency strength of the resulting theory?