On the Principles of Reflection in Axiomatic Set Theory

Abstract

Publisher Summary This chapter discusses the principles of reflection of axiomatic set theory. The chapter describes the set theory that is formalized in the first-order to predicate calculus with equality and with the binary predicate symbol ɛ, as its only non logical constant. The chapter also indicates S as the set theory consisting of the axioms of extensionality, pairing, union-set, power-set, and the axiom schemata of subsets and foundation. It also includes the Zermelo–Fraenkel set theory (ZF) as the set theory obtained from S by addition of the axiom of infinity and the axiom schema of replacement. All the theorems in this chapter are theorems in respective axiomatic set theories. The theory in which the theorem is proved can be explicitly indicated for each numbered theorem. The chapter uses the theorem schema of S on definitions by recursion that asserts that a term F(α) can be defined by recursion so that F(α) satisfies the recursive condition, if α is such that there is a set containing all ordered pairs for β , and F(α) is 0 otherwise.