On Induction vs. *-Continuity

In this paper we study the relative expressibility of the infinitary *-continuity condition $$ X \equiv V_n X$$ (*-cont) and the equational but weaker induction axiom $$X \wedge [\alpha ^* ](X \supset [\alpha ]X) \equiv [\alpha ^* ]X$$ (ind) in Propositional Dynamic Logic. We show: (1) under ind only, there is a first-order sentence distinguishing separable dynamic algebras from standard Kripke models; whereas (2) under the stronger axiom *-cont, the class of separable dynamic algebras and the class of standard Kripke models are indistinguishable by any sentence of infinitary first-order logic.