Abstract Sets and Finite Ordinals [electronic resource] : An Introduction to the Study of Set Theory

This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision. 1961 edition. This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision. Starting with an explanation of all the basic logical terms and related operations, the text progresses through a stage-by-stage elaboration that proves the fundamental theorems of finite sets. It focuses on the Bernays theory of finite classes and finite sets, exploring the system's basis and development, including Stage I and Stage II theorems, the theory of finite ordinals, and the theory of finite classes and finite sets. This volume represents an excellent text for undergraduates studying intermediate or advanced logic as well as a fine reference for professional mathematicians. 1.1. Primitive Constants1.2. Initial Definitions; 1.3. The Axioms of the System; 1.4. The Classes Admitted by AxIII; SECTION - DEVELOPMENT OF THE SYSTEM: STAGE I; 2.1. Stage I Theorems; 2.2. Standard Expressions and the Class Theorem; SECTION 3 - DEVELOPMENT OF THE SYSTEM: STAGE II; 3.1. Stage II Theorems; 3.2. Fundamental Theorems on Ordinals; SECTION 4 - THE THEORY OF FINITE ORDINALS; 4.1. Finite Ordinal Number; 4.2. The Peano Axioms as Theorems of the System; 4.3. Iteration; 4.4. The Existence Theorem for the Function : . (ET); 4.5. Addition, Multiplication, and Exponentiation Title Page; Dedication; Copyright Page; Table of Contents; PREFACE; PART I - THE ELEMENTS OF SET THEORY; SECTION 1 - THE BASIC LOGICAL CONCEPTS; 1.1. Introduction; 1.2. The Logical Constants; 1.3 The Use of Quantifiers in Proofs; SECTION 2 - OPERATIONS ON CLASSES; 2.1. Classes and Composite Wholes; 2.2. Conventions for Diagramxning Class-structure; 2.3. Elementary Operations; PART II - THE BERNAYS THEORY OF FINITE CLASSES AND FINITE SETS; FOREWORD TO PART TWO; INTRODUCTION; 0.1. Pairs, Ordered Pairs and k-Tuplets; 0.2. Class Conditions; SECTION 1 - THE BASIS OF THE SYSTEM SECTION 5 - THE THEORY OF FINITE CLASSES AND FINITE SETS5.1. Preliminary Theorems on Finite Ordinals; 5.2. Theorems on Finite Classes and Sets; 5.3 Conclusion; REFERENCES; INDEX