From the Preface:
of mathematical logic be put forth. As the papers deal with only a few subdomains of the field, I have added from unpublished notes discussions of a number of standard topics and replaced a few papers by somewhat more general descriptions of related matter. The result is the present book.
As a survey, it is neither complete nor impartial. For example, modal logic, combinatory logic, and many-valued logic are not considered at all. More serious omissions are any adequate account of more recent works on recursive functions and on intuitionism, two of the subjects which are of the greatest current interest among working mathematical logicians. But these two fields are in the midst of rapid developments; there seems to be no choice but to follow new discoveries as they come out. In a different direction, works not easily accessible to me have not been adequately dealt with.
Author(s): Hao Wang
Publisher: Science Press / North-Holland Publishing Company
Year: 1963
Language: English
Commentary: Added pp. 100-101.
Pages: 663
Cover
Title Page
Preface
Contents
Part One: General Sketches
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
References
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
III. On Formalization
§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
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
References
V. Computation
§1. The Concept of Computability
§2. General Recursive Functions
§3. The Friedberg-Mucnik Theorem
§4. Metamathematics
§6. The Control of Errors in Calculating Machines
References
Part Two: Calculating Machines
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
References
VII. Universal Turing Machines: An Exercise in Coding
References
VIII. The Logic of Automata
§1. Introduction
§2. Automata and Nets
2.1. Fixed and Growing Automata
2.2. Characterizing Tables and a Decision Procedure
2.3. Representation by Nets. The Coded Normal Form
§3. Transition Matrices and Matrix Form Nets
3.1. Transition Matrices
3.2. Matrix Form Nets
3.3. Some Uses of Matrices
§4. Cycles, Nets, and Quantifiers
4.1. Decomposing Nets
4.2. Truth Functions and Quantifiers
4.3. Nerve Nets
References
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 Systems 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
Appendix I: A sample from print-outs by Program I
Appendix II: Sample from print-outs by Program II
Appendix III: Sample of print-outs by Program III (no quantifiers)
Appendix IV: Sample of print-outs by Program III (the AE predicate calculus)
Appendix V: Different methods for the AE predicate calculus
Appendix VI: An example of Church’s
Appendix VII: An example of Quine
References
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
References
Part Three: Formal Number Theory
XI. The Predicate Calculus
§1. The Propositional Calculus
§2. Formulations of the Predicate Calculus
§3. Completeness of the Predicate Calculus
XII. Many-Sorted Predicate Calculus
§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)
XIII. The Arithmetization of Metamathematics
§1. Gödel Numbering
§2. Recursive Functions and the System Z
§3. Bernays’ Lemma
§4. Arithmetic Translations of Axiom Systems
XIV. Ackermann’s Consistency Proof
§1. The System Z_a
§2. Proof of Finiteness
§3. Estimates of the Substituents
§4. Interpretation of Nonfinitist Proofs
XV. Partial Systems of Number Theory
§1. Skolem’s Non-standard Model for Number Theory
§2. Some Applications of Formalized Consistency Proofs
Part Four: Impredicative Set Theory
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
References
XVII. Relative Strength and Reducibility
§1. Relation between P and Q
§2. Finite Axiomatization
§3. Finite Sets and Natural Numbers
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
References
XIX. Between Number Theory and Set Theory
§1. General Set Theory
§2. Predicative Set Theory
§3. Impredicative Collections and ω-Consistency
References
XX. Some Partial Systems
§1. Some Formal Details on Class Axioms
§2. A New Theory of Element and Number
2.1. Basal Logic and Elementhood Axioms
2.2. The Arithmetic of Natural Numbers
2.3. Model and Enumeration of Elements
§3. Set-Theoretical Basis for Real Numbers
3.1. Introduction
3.2. The System L
3.3. Real Numbers
§4. Functions of Real Variables
References
Part Five: Predicative Set Theory
XXI. Certain Predicates Defined by Induction Schemata
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
References
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 Limitation 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
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. Proofs of Consistency
§10. The System R
10.1. Levels, Types, Orders
10.2. Expressions of the System R
10.3. Fundamental Definitions and Axioms of R
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
References
