Some impredicative definitions in the axiomatic set theory

Abstract

Publisher Summary This chapter provides some impredicative definitions in the set-theory. Zermelo–Fraenkel (ZF) set-theory is denoted by S . The chapter assumes the well-known rules of proof—namely, the modus ponens, the rule of substitution and the rules of omission and of introduction of quantifiers. The chapter assumes special rules that enable to prove every tautological formula including the identity-symbol. S' is assumed to be the Bernays–Godel system of set-theory. Every expression meaningful in ( S ) is also meaningful in ( S' ) and every axiom of ( S ) is provable in ( S' ). The chapter describes two different ways to express in ( S' ) meta-mathematical theorems about ( S ). One of them uses the method of Godel and identifies expressions with their Godel numbers. Another possible method is to express theorems about formulas of ( S ) as theorem schemata of ( S' ). It is demonstrated that proofs of all theorems and of all particular instances of the schemata are based exclusively on the axioms of ( S' ).