A Survey of Mathematical Logic

Table of contents :
Title
Copyright
Preface
Contents
PART 1. GENERAL SKETCHES
Chapter I. The axiomatic method
1. Geometry and axiomatic systems
2. The problem of adequacy
3. The problem of evidence
4. A very elementary system L
5. The theory of non-negative integers
6. Gödel's theorems
7. Formal theories as applied elementary logics
Chapter II. Eighty years of foundational studies
1. Analysis, reduction and formalization
2. Anthropologism
3. Finitism
4. Intuitionism
5. Predicativism: standard results on number as being
6. Predicativism: predicative analysis and beyond
7. Platonism
8. Logic in the narrower sense
9. Applications
Chapter III. On formalizatlon
1. Systematization
2. Communication
3. Clarity and consolidation
4. Rigour
5. Approximation to intuition
6. Application to philosophy
7. Too many digits
8. Ideal language
9. How artificial a language?
10. The paradoxes
Chapter IV. The axiomatization of arithmetic
1. Introduction
2. Grassmann's calculus
3. Dedekind' s letter
4. Dedekind's essay
5. Adequacy of Dedekind's characterization
6. Dedekind and Frege
Chapter V. Computation
1. The concept of computability
2. General recursive functions
3. The Friedberg-Mucnik theorem
4. Metamathematics
5. Symbolic logic and calculating machines
6. The control of errors in calculating machines
PART 2. CALCULATING MACHINES
Chapter VI. A variant to Turing's theory of calculating machines
1. Introduction
2. The basic machine B
3. All recursive functions are B-computable
4. Basic instructions
5. Universal Turing machines
6. Theorem-proving machines
Chapter VII. Universal Turing machines: an exercise in coding
Chapter VIII. The logic of automata (with A. W. Burks)
1. Introduction
2. Automata and nets
3. Transition matrices and matrix form nets
4. Cycles, nets, and quantifiers
Chapter IX. Toward mechanical mathematics
1. Introduction
2. The propositional calculus (system P)
3. Program I: the propositional calculus P
4. Program II: selecting theorems in the propositional calculus
5. Completeness and consistency of the system P and P_s
6. The system P_e: the propositional calculus with equality
7. Preliminaries to the predicate calculus
8. The system Q_p and the AE predicate calculus
9. Program III
10. Systems Q_q and Q_r: alternative formulations of the AE predicate calculus
11. System Q: the whole predicate calculus with equality
12. Conclusions
Appendices I-VII
Chapter X. Circuit synthesis by solving sequential Boolean equations
1. Summary of problems and results
2. Sequential Boolean functionals and equations
3. The method of sequential tables
4. Deterministic solutions
5. Related problems
6. An effective criterion of general solvability
7. A sufficient condition for effective solvability
8. An effective criterion of effective solvability
9. The normal form (S) of sequential Boolean equations
10. Apparently richer languages
11. Turing machines and growing automata
PART 3. FORMAL NUMBER THEORY
Chapter XI. The predicate calculus
1. The propositional calculus
2. Formulations of the predicate calculus
3. Completeness of the predicate calculus
Chapter XII. Many-sorted predicate calculi
1. One-sorted and many-sorted theories
2. The many-sorted elementary logics L_n
3. The theorem (I) and the completeness of L_n
4. Proof of the theorem (IV)
Chapter XIII. The arithmetization of metamathematics
1. Gödel numbering
2. Recursive functions and the system Z
3. Bernays' lemma
4. Arithmetic translations of axiom systems
Chapter XIV. Ackermann's consistency proof
1. The system Z_a
2. Proof of finiteness
3. Estimates of the substituents
4. Interpretation of nonfinitist proofs
Chapter XV. Partial systems of number theory
1. Skolem's non-standard model for number theory
2. Some applications of formalized consistency proofs
PART 4. IMPREDICATIVE SET THEORY
Chapter XVI. Different axiom systems
1. The paradoxes
2. Zermelo's set theory
3. The Bernays set theory
4. The theory of types, negative types, and "new foundations"
5. A formal system of logic
6. The systems of Ackermann and Frege
Chapter XVII. Relative strength and reducibility
1. Relation between P and Q
2. Finite axiomatization
3. Finite sets and natural numbers
Chapter XVIII. Truth definitions and consistency proofs
1. Introduction
2. A truth definition for Zermelo set theory
3. Remarks on the construction of truth definitions in general
4. Consistency proofs via truth definitions
5. Relativity of number theory and in particular of induction
6. Explanatory remarks
Chapter XIX. Between number theory and set theory
1. General set theory
2. Predicative set theory
3. Impredicative collections and ω-consistency
Chapter XX. Some partial systems
1. Some formal details on class axioms
2. A new theory of element and number
3. Set-theoretical basis for real numbers
4. Functions of real variables
PART 5. PREDICATIVE SET THEORY
Chapter XXI. Certain predicates defined by induction schemata
Chapter XXII. Undecidable sentences suggested by semantic paradoxes
1. Introduction
2. Preliminaries
3. Conditions which the set theory is to satisfy
4. The Epimenides paradox
5. The Richard paradox
6. Final remarks
Chapter XXIII. The formalization of mathematics
1. Original sin of the formal logician
2. Historical perspective
3. What is a set?
4. The indenumerable and the impredicative
5. The limitations upon formalization
6. A constructive theory
7. The denumerability of all sets
8. Consistency and adequacy
9. The axiom of reducibility
10. The vicious-circle principle
11. Predicative sets and constructive ordinals
12. Concluding remarks
Chapter XXIV. Some formal details on predicative set theories
1. The underlying logic
2. The axioms of the theory Σ
3. Preliminary considerations
4. The theory of non-negative integers
5. The enumerability of all sets
6. Consequences of the enumerations
7. The theory of real numbers
8. Intuitive models
9. Praofs of consistency
10. The system R
Chapter XXV. Ordinal numbers and predicative set theory
1. Systems of notation for ordinal numbers
2. Strongly effective systems
3. The Church-Kleene class B and a new class C
4. Partial Herbrand recursive functions
5. Predicative set theory
6. Two tentative definitions of predicative sets
7. System H: the hyperarithmetic set theory