The Foundations of Mathematics 9781904987147
789 112 2MB
English Pages 262 Year 2011
Table of contents :
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
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