Two Contributions to the Foundations of Set Theory

Abstract

A simple alternative formulation is given in the chapter for an axiom that is recently proposed by Lévy as an addition to the axioms of Zermelo–Fraenkel set theory. If T is a theory obtained from Zermelo set theory by adding all instances with no more than a given number of alternations of quantifiers of a schema valid in Zermelo–Fraenkel set theory, then the consistency of T is demonstrable in Zermelo–Fraenkel set theory (a slightly stronger result is obtained). The chapter also mentions the term expressions that are to be identified with natural numbers. The chapter distinguishes an infinite set of natural numbers that are to be known as variables and—for each positive integer n—an infinite set of natural numbers to be known as n place predicate.