The Foundations of Mathematics

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Cover Title Copyright Contents Preface Chapter 0 Introduction 0.1 Prerequisites 0.2 Predicate Logic 0.3 Why Read This Book? 0.4 The Foundations of Mathematics 0.5 How to Read This Book Chapter I Set Theory I.1 Plan I.2 The Axioms I.3 Two Remarks on Presentation I.4 Set theory is the the theory of everything I.5 Counting I.6 Extensionality, Comprehension, Pairing, Union I.7 Relations, Functions, Discrete Mathematics I.7.1 Basics I.7.2 Foundational Remarks I.7.3 Well-orderings I.8 Ordinals I.9 Induction and Recursion on the Ordinals I.10 Power Sets I.11 Cardinals I.12 The Axiom of Choice (AC) I.13 Cardinal Arithmetic I.14 The Axiom of Foundation I.15 Real Numbers and Symbolic Entities Chapter II Model Theory and Proof Theory II.1 Plan II.2 Historical Introduction to Proof Theory II.3 NON-Historical Introduction to Model Theory II.4 Polish Notation II.5 First-Order Logic Syntax II.6 Abbreviations II.7 First-Order Logic Semantics II.8 Further Semantic Notions II.9 Tautologies II.10 Formal Proofs II.11 Some Strategies for Constructing Proofs II.12 The Completeness Theorem II.13 Complete Theories II.14 Equational and Horn Theories II.15 Extensions by Definitions II.16 Elementary Submodels II.17 Definability and Absoluteness in Models of Set Theory II.18 Some Weaker Set Theories II.19 Other Proof Theories Chapter III The Philosophy of Mathematics III.1 What Is Really True? III.2 Keeping Them Honest III.3 On the EI Rule and AC Chapter IV Recursion Theory IV.1 Overview IV.2 The Church-Turing Thesis IV.3 Δ1 relations on HF IV.4 Diagonal Arguments IV.5 Decidability in Logic Bibliography Index Back Cover