Mathematical Logic and The Foundations of Mathematics

Preface
Acknowledgements
Contents
Part I MATHEMATICAL LOGIC
	Chapter 1 Traditional Logic
		1 Introduction: mathematics and logic
		2 The nature of mathematics
		3 The nature of logic
		4 Terms and propositions
		5 Syllogistic inference
		6 The reduction of syllogisms
		7 Deductive arguments of more complex form
		SUPPLEMENTARY NOTES ON CHAPTER 1
			1 Aristotle's 'Organon'
			2 Transitional logic
			3 Venn's diagrams
			4 Additional reading
			5 'The Development of Logic'
	Chapter 2 Symbolic Logic I: The Propositional Calculus
		1 Propositional logic
		2 Prepositional variables and the basic connectives
		3 Equiveridicity of formulae
		4 Elementary transformations in the propositional calculus
		5 Normal forms
		6 Axiomatic treatment of the propositional calculus
		SUPPLEMENTARY NOTES ON CHAPTER 2
			1 The term 'semantic'
			2 The logical notation of Lukasiewicz
			3 The sixteen truth-functions of two propositional variables
			4 Boole's application of algebraic symbolism to logic
			5 Logical sum and logical product
			6 C. I. Lewis's modal logic of strict implication
			7 Many-valued logics
			8 Books on symbolic logic
	Chapter 3 Symbolic Logic II: The Restricted Calculus of Predicates
		1 Prepositional functions
		2 Quantification
		3 Axiomatic treatment of the calculus of predicates
		4 Completeness of the restricted calculus of predicates
		5 The deduction theorem
		SUPPLEMENTARY NOTES ON CHAPTER 3
			1 Some points concerning symbolism
			2 A strong sense of 'complete'
	Chapter 4 Further Development of Symbolic Logic
		1 The relation of identity
		2 Descriptions and the ℩-symbol
			2.1 Characteristic functions of formulae
		3 Formalized mathematical theories
		4 Hilbert's ε-symbol
			4.1 Symbolic resolution of existential axioms
		5 Classes and relations
		6 The extended calculus of predicates
			6.1 Russell's antinomy
			6.2 Axiomatization of the extended calculus of predicates
		7 The logical calculus of Bourbaki's 'Éléments de Mathématique'
		8* Gentzen's calculus of natural deduction
		SUPPLEMENTARY NOTES ON CHAPTER 4
			1 The vicious-circle paradoxes
			2 Tarski's formalization of semantics
			3 The ε-symbol as a selection operator
			4 Additional reading
Part II FOUNDATIONS OF MATHEMATICS
	Chapter 5 The Critical Movement in Mathematics in the Nineteenth Century
		1 Symbolic logic in relation to the foundations of mathematics
		2 Greek mathematics
		3 The beginnings of modern mathematics
		4 The first phase of the critical movement
		5 Peano's 'Formulaire de Mathématique'
		6 The symbolic language of Leibniz
		SUPPLEMENTARY NOTES ON CHAPTER 5
			1 Peano's space-filling curve
			2 Ideal numbers and ideals
			3 Sources of historical information
	Chapter 6 The Logistic Identification of Mathematics with Logic
		1 Russell's conception of mathematics
		2 Dedekind's analysis of number
			2.1 Cantor's definition of cardinal and ordinal number
		3 'Principia Mathematica'
			3.1 The logical calculus of 'Principia Mathematica'
			3.2 Russell's theory of types
			3.3 The formal development of 'Principia Mathematica'
		4 Frege's logical analysis of arithmetic
			4.1* The 'Begriffsschrift'
			4.2* The 'Grundgesetze der Arithmetik'
		SUPPLEMENTARY NOTES ON CHAPTER 6
			1 Non-euclidean geometry
			2 Sense and denotation
			3 The algebra of logic
			4 '⊢' as a symbol for derivability
			5 Introductory books on the foundations of mathematics
			6 Books on the logistic conception of mathematics
			7 Sources
	Chapter 7 Formalized Mathematics and Metamathematics
		1 Hilbert's new approach to the foundations of mathematics
		2 Hilbert's 'Grundlagen der Geometrie'
		3 Axiomatic theories and their significance
		4 The domain of numerals, treated by finitary means
		5 The metama thematics of formalized theories
		6 Consistency of the restricted calculus of predicates
		7 Consistency of arithmetic
		SUPPLEMENTARY NOTES ON CHAPTER 7
			1 Foundations of projective geometry
			2 Models of non-euclidean geometry
			3 The literature of Hilbert's metamathematics
	Chapter 8 Gödel's Theorems on the Inherent Limitations of Formal Systems
		1 Gödel's new metamathematical method
		2 Gödel's heuristic argument
		3 The formal system F and its arithmetized metamathematics
		4* The central argument of Gödel's paper
		5 The impossibility of an 'internal' proof of consistency
		SUPPLEMENTARY NOTES ON CHAPTER 8
			1 Gentzen's proof of the consistency of arithmetic
			2 Alternative accounts of Gödel's investigation
	Chapter 9 Intuitionism
		1 The intuitionist outlook
		2 Primary intuition
		3* Intuitionist analysis
		4* Heyting's formalization of intuitionist logic
		SUPPLEMENTARY NOTES ON CHAPTER 9
			1 Accounts of Brouwer's intuitionism
			2 Weyl's intuitionism
	Chapter 10 Recursive Arithmetic
		1 The class of natural numbers as a progression
		2 Recursive definitions
		3 Systematic development of recursive arithmetic
		4 Recursive arithmetic as a formal system
		5 Representability of recursive arithmetic in the system (Z)
		6 Primitive recursive and other recursive schemata
		7 General recursive functions
		8 The mathematical ideal of constructivity
			8.1 Church's calculus of λ-conversion
			8.2 Turing's conception of computability
			8.3 The decision problem for formalized theories
			8.4* Post's canonical form for formal systems
		SUPPLEMENTARY NOTES ON CHAPTER 10
			1 Books on recursive arithmetic
			2 Books on constructivity in general
	Chapter 11 The Axiomatic Theory of Sets
		1 Pure mathematics as an extension of the theory of sets
		2 The naive theory of sets
		3 Zermelo's axiomatic theory of sets
		4 Von Neumann's new approach to the theory of sets
		5 Bernays's unification of symbolic logic and the theory of sets
		6 The theory of sets in Bourbaki's 'Éléments de Mathématique'
		7 Limitations of the axiomatic treatment of sets
		SUPPLEMENTARY NOTES ON CHAPTER 11
			1 Historically important papers on the theory of sets
			2 Additional reading
Part III PHILOSOPHY OF MATHEMATICS
Chapter 12 The Epistemological Status of Mathematics
	1 Retrospect
	2 The logistic conception of mathematics
	2.1 Mathematics as an edifice of propositions
	2.2 Logic and objective reality
	3 The relativism of Brouwer's intuitionist outlook
	4 Hilbert's twenty-three mathematical problems
	5 Bourbaki's interpretation of mathematics
Chapter 13 The Application of Mathematics to the Natural World
	1 Pure mathematics and applied mathematics
	2 The spatial structure of the world
	3 Naive realism and its inadequacy
	4 The a priori form of the physical world
	5 Whitehead's theory of natural knowledge
		5.1 The method of extensive abstraction
		5.2 Definition of time and space by extensive abstraction
		5.3 Objects
	6 Mathematics and the logical analysis of the natural world
	SUPPLEMENTARY NOTES ON CHAPTER 13
		1 Zeno's paradoxes of motion
		2 The theory of games
		3 The mathematical theory of crystalline structure
		4 Books on the philosophy of science
Chapter 14 Logic and the Activity of Thinking
	1 The limitations of formal logic
	2 The logic of concrete thought
	3 Inductive reasoning
		3.1 The element of judgement in induction
		3.2 The calculus of probabilities
	4 The genetic method in philosophy
	5 The outcome of the philosophy of mathematics
Appendix Developments since 1939 in the Study of Foundations of Mathematics
	1 Mathematical logic
	2 Consistency of analysis
	3 Constructive treatment of ordinal numbers
	4 Modern views on proof of consistency
	5 Models
	6 Decision procedures
	7 Effectiveness and constructivity
	8 Intuitionism
	9 The theory of sets
	10 Many-valued logics
	11 Metamathematics and algebra
	12 Modal logic
	13 Formalization of semantics
	14 Probability and induction
	15 Nominalism and platonism
	16 Logic and computing machines
Bibliography
The Greek and German Alphabets
Index of Symbols
General Index