What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs?Only in this century has there been success in obtaining substantial and satisfactory answers。 The present book contains a systematic discussion of these results。 The investigations are centered around first-order logic。 Our first goal is’ Godel’s completeness theorem, which shows that the consequence relation coincides with formal provability

Preface
PART A
ⅠIntroduction
1.An Example from Group Theory
2.An Example from the Theory of Equivalence Relations
3.A Preliminary Analysis
4.Preview
Ⅱ Syntax of First-Order Languages
1.Alphabets
2.The Alphabet of a First-Order Language
3.Terms and Formulas in First-Order Languages
4.Induction in the Calculus of Terms and in the Calculus of Formulas
5.Free Variables and Sentences
Ⅲ Semantics of First-Order Languages
1.Structures and Interpretations
2.Standardization of Connectives
3.The Satisfaction Relation
4.The Consequence Relation
5.Two Lemmas on the Satisfaction Relation
6.Some simple formalizations
7.Some remarks on Formalizability
8.Substitution
Ⅳ A Sequent Calculus
1.Sequent Rules
2.Structural Rules and Connective Rules
3.Derivable Connective Rules
4.Quantifier and Equality Rules
5.Further Derivable Rules and Sequents
6.Summary and Example
7.Consistency
ⅤThe Completeness Theorem
1.Henkin’S Theorem.
2. Satisfiability of Consistent Sets of Formulas(the Countable Casel
3. Satisfiability of Consistent Sets of Formulas(the General Case)
4.The Completeness Theorem
Ⅵ The LSwenheim-Skolem and the Compactness Theorem
1.The L6wenheim-Skolem Theorem.
2.The Compactness Theorem
3.Elementary Classes
4.Elementarily Equivalent Structures
Ⅶ The Scope of First-Order Logic
1.The Notion of Formal Proof
2.Mathematics Within the Framework of Fimt—Order Logic
3.The Zermelo-Fraenkel Axioms for Set Theory.
4.Set Theory as a Basis for Mathematics
Ⅷ Syntactic Interpretations and Normal Forms
1.Term-Reduced Formulas and Relational Symbol Sets
2.Syntactic Interpretations
3.Extensions by Definitions
4.Normal Forms
PART B
Ⅸ Extensions of First-order logic
Ⅹ Limitations of the Formal Method
Ⅺ Free Models and Logic Programming
Ⅻ An Algebraic Characterization of Elementary Equivalence
ⅩⅢ Lindstrom’s Theorems
References
Symbol Index
Subject Index