An equivalent form of Lévy’s axiom schema

A generalized notion of ordinal inaccessibility is de- fined. A characterization of this notion in terms of normal ordinal functions yields as a consequence that a schema analogous to a form of Tarski's Axiom of Inaccessible Cardinals is equivalent to Levy's axiom schema unider the axiom of choice. This note supplies proof to the result announced in (I). Using ele- mentary ordinal theory alone we derive a schema that is equivalent to that of Levy in the presence of AC (the axiom of choice) and that has a simple relationship to Tarski's Axiom of Inaccessible Cardinals. R. Montague has obtained another equivalent form of Levy's schema as a consequence of a model-theoretic lemma proved in Montague (3). It turns out that our form and Montague's form of the Levy schema correspond to the two commonly stated forms of Tarski's Axiom of Inaccessible Cardinals. We mnake use of the notation and definitions of Levy (2), in particular we borrow the illegitimate con- venience of speaking of functions with domain the class of all ordinals. Actually, all our "functions" will be of this kind and instead of or- dinal-valued function defined for all ordinals we will write simply func- tion. A canonical example is the ordinal successor function. By weakly (strongly) inaccessible we shall mean im weiteren (im engeren) Sinne unerreichbar in the sense of (4), but inaccessible (unqualified) has the sense of (2 ). 1. Three schemata. Consider the following three statements: