AXIOM SCHEMATA OF STRONG INFINITY IN AXIOMATIC SET THEORY

l Introduction. There are, in general, two main approaches to the introduction of strong infinity assertions to the Zermelo-Fraenkel set theory. The arithmetical approach starts with the regular ordinal numbers, continues with the weakly inaccessible numbers and goes on to the ^-numbers of Mahlo [4], etc. The model-theoretic approach, with which we shall be concerned, introduces the strongly inaccessible numbers and leads to Tarski's axioms of [14] and [15]. As we shall see, even in the model-theoretic approach we can use methods for expressing strong assertions of infinity which are mainly arithmetical. Therefore we shall introduce strong axiom schemata of infinity by following Mahlo [4,5,6,]. Using the ideas of Montague in [7] we shall give those axiom schemata a purely model-theoretic form. Also the axiom schemata of replacement in conjunction with the axiom of infinity will be given a similar form, and thus the new axiom schemata will be seen to be natural continuations of the axiom schema of replacement and infinity. A provisional notion of a standard model, introduced in § 2, will be basic for our discussion. However, in § 5 it is shown that this definition cannot serve as a general definition for the notion of a standard model.