A Study on Axioms and Models of Zermelo-Fraenkelset Theory

Abstract

Background: Sets are given axiomatically, thus their existence and basic properties are asserted by appropriate formal axioms. Axiomatic systems address the inconsistencies associated with naïve set theory. The most commonly used system of axioms for sets is the Zermelo-Fraenkel set theory (ZF).  Objectives: In this work, we intend to investigate some set-theoretic results. We begin with a study on the standard axioms of Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC). A comparative analysis of the von Neumann-Bernays-Godel (NBG), which is a conservative extension of ZFC, is presented. Methods: Axiomatic set theory is developed in the framework of the first-order predicate calculus; this allows a formalization of all mathematical notions and arguments. A model of the Zermelo-Fraenkel set theory that is obtained through an iterative construction that follows the von Neumann hierarchy is presented. Results: The axioms of Zermelo-Fraenkel set theory and the axiom of choice ZFC were discussed in this work. A comparative analysis of the von Neumann-Bernays-Goedel set theory, which is a conservative extension of ZFC, was presented. The iterative conception of set proposed by von Neumann is a model of set theory and by varying the technique of recursion on the ordinals, different graph models of set theory can be constructed. Conclusions: These graph models can be employed in establishing independence and consistency results in set theory.

Keywords: Axiomatic system, Formal language, Models, First-order logic and von Neumann hierarchy