Introduction

Abstract

Leafing through almost any exposition of modern mathematical logic, including this book, one will note the highly technical and purely mathematical nature of most of the material. Generally speaking this may seem strange to the novice, who pictures logic as forming the foundation of mathematics and expects to find many difficult discussions concerning the philosophy of mathematics. Even more puzzling to such a person is the fact that most works on logic presuppose a substantial amount of mathematical background, in fact, usually more set theory than is required for other mathematical subjects at a comparable level. To the novice it would seem more appropriate to begin by assuming nothing more than a general cultural background. In this introduction we want to try to justify the approach used in this book and similar ones. Inevitably this will require a discussion of the philosophy of mathematics. We cannot do full justice to this topic here, and the interested reader will have to study further, for example in the references given at the end of this introduction. We should emphasize at the outset that the various possible philosophical viewpoints concerning the nature or purpose of mathematics do not effect one way or the other the correctness of mathematical reasoning (including the technical results of this book). They do effect how mathematical results are to be intuitively interpreted, and which mathematical problems are considered as more significant.