The investigations of Andrzej Mostowski in the foundations of set theory

Abstract

Publisher Summary This chapter focuses on two problems, which Andrzej Mostowski faced in investigating the foundations of set theory: the systems of axioms of set theory and the logical relations between various sentences derivable from those systems. Mostowski's work concerned two essential types of theories: the Zermelo–Fraenkel theories with or without individuals, and second order theories of the Godel–Bernays type and of the Kelley–Morse type. Mostowski's doctoral dissertation was devoted to various forms of the definition of infinity. The set theory with individuals known today as the Fraenkel–Mostowski system was formulated by Mostowski in paper 6. In Mostowski's investigations a particular role is played by the problem of the independence of the axiom of choice of other set-theoretical axioms and the problem of the independence of various forms of the axiom of choice. In 1951, Mostowski investigated the relations between the Zermelo–Fraenkel set theory and the Godel –Bernays theory of classes.