Foundations of Mathematics in the Twentieth Century

Abstract

1. INTRODUCTION AND EARLY DEVELOPMENTS. Logic and foundations are a domain of mathematics concerned with basic mathematical structures (in terms of which one can define all other mathematical structures), with the correctness and significance of mathematical reasoning, and with the effectiveness of mathematical computations. In the twentieth century, these areas have crystallized into three large chapters of mathematics: set theory, mathematical logic (including model theory), and computability theory, which are intertwined in significant ways. In this paper we describe the evolution and present state of each of them. In modern times the study of logic and foundations has attracted eminent mathematicians and philosophers such as Cantor, Cohen, Frege, Godel, Hilbert, Kleene, Martin, Russell, Solovay, Shelah, Skolem, Tarski, Turing, Zermelo, and others, and has given rise to a large body of knowledge. Although our paper is only a brief sketch of this development, we discuss essential results such as G6del's theorem on the completeness of first-order logic and his theorems on the incompleteness of most mathematical theories, some independence theorems in set theory, the role of axioms of existence of large cardinal numbers, Turing's work on computability, and some recent developments. There are still many interesting unsolved problems in logic and foundations. For example, logic does not explain what mathematics is. We know that mathematics has a very precise structure: axioms, definitions, theorems, proofs. Thus we know what is correct mathematics but not why the works of certain mathematicians delight us while others strike us as downright boring. Nor do foundations tell us how mathematicians construct proofs of their conjectures. Since we have no good theoretical model of the process for constructing proofs, we are far from having truly effective procedures for automatically proving theorems, although spectacular successes have been achieved in this area. We mention other unsolved problems at the end of this paper. We now give a brief sketch of the history of logic and foundations prior to 1900. The ancient Greeks asked: what are correct arguments? and: what are the real numbers? As partial answers they created the theory of syllogisms and a theory of commensurable and incommensurable magnitudes. These questions resurfaced in the 18th century, due to the development of analysis and to the lack of sufficiently clear concepts of sets, functions, continuity, convergence, etc. In a series of papers (1878-1897) Georg Cantor created set theory. In 1879 Gottlob Frege described a formal system of logic that explained precisely the logical structure of all mathematical proofs. In 1858 Richard Dedekind gave a definitive answer to the question: what are the real numbers? by defining them in terms of sets of rational numbers. He proved the axiom of continuity of the real line, an axiom accepted hitherto (beginning with the Greeks) without proof.