The Foundations of Mathematics in Poland After World War II

Abstract

Publisher Summary This chapter focuses on set theory, model theory, recursion theory and their foundation. The chapter investigates the equivalents of continuum hypothesis and presents a general method of independence proofs in set theory with atoms. This method, which is currently called the Fraenkel–Mostowski method, is based on a set of individuals—that is, objects that are not sets. The chapter presents the results of the investigations in the foundations of set theory. A result with far-reaching consequences is a collection of facts on seemingly paradoxical properties of the Godel–Bernays theory of classes. One of the basic tools for modern investigations in the foundations of set theory is “Mostowski's contraction lemma.” This lemma is a generalization of the fact that every well-ordering is similar to some von Neumann ordinal. The chapter further discusses the formulation of the so-called Kelley–Morse theory of classes. The set-theoretical objects are of two possible kinds: the “small” ones, which belong to another object (these are sets), and “large” ones, which do not belong to any object (these are proper classes).