.

0 (hodnocen0 x )

(2) Půjčeno:2x 

BK

2nd ed.

New York : Springer, c1994

x, 289 s.

ISBN 0-387-94258-0 (váz.)

Undergraduate texts in mathematics

Obsahuje rejstřík.

Bibliografie: s. 277-279.

Logika matematická - učebnice vysokošk.

000058626

Preface // PART A // I Introduction // 3 // §1. An Example from Group Theory // An Example from the Theory of Equivalence Relations . . . // §3. A Preliminary Analysis // §4. Preview // II Syntax of First-Order Languages // §1. Alphabets...11 // §2. The Alphabet of a First-Order Language...13 // s and For // §4. Induction in the Calculus of Terms and in the Calculus of // Formulas ...19 // §5. Free Variables and Sentences...24 // III Semantics of First-Order Languages ...27 // §1. Structures and Interpretations...28 // §2. Standardization of Connectives...31 // §3. The Satisfaction Relation...32 // §4. The Consequence Relation...33 // §5. Two Lemmas on the Satisfaction Relation...40 // §6. Some Simple Formalizations...44 // §7. Some Remarks on Formalizability...48 // §8. Substitution...52 // viii Contents // IV A Sequent Calculus 59 // §1. Sequent Rules...60 // §2. Structural Rules and Connective Rules...62 // §3. Derivable Connective Rules...63 // §4. Quantifier and Equality Rules...66 // §5. Further Derivable Rules and Sequents...68 // §6. Summary and Example ...69 // §7. Consistency...72 // V The Completeness Theorem ..75 // §1. Henkin’s Theorem...75 // §2. Satisfiability of Consistent Sets of Formulas (the Countable Case) ...79 // §3. Satisfiability of Consistent Sets of Formulas (the General Case) 82 // The Completeness Theorem // VI The Löwenheim-Skolem and the Compactness Theorem 87 // The Löwenheim-Skolem Theorem ... 87 // §2. The Compactness Theorem...88 // §3. Elementary Classes...91 // §4. Elementarily Equivalent Structures...94 // VII The // Scope of First-Order Logic ...99 // §1. The Notion of Formal Proof...99 // §2. Mathematics Within the Framework of First-Order Logic . 103 // §3. The Zermelo-Fraenkel Axioms for Set Theory...107 // §4. Set Theory as a Basis for Mathematics...110 //

VIII Syntactic Interpretations and Normal Forms 115 // §1. Term-Reduced Formulas and Relational Symbol Sets ...115 // §2. Syntactic Interpretations...118 // §3. Extensions by Definitions...125 // §4. Normal Forms...128 // PART B // IX Extensions of First-Order Logic // 135 // 137 // §1. Second-Order Logic...138 // §2. The System C,u...142 // §3. The System Cq...148 // X Limitations of the Formal Method // 151 // §1. Decidability and Enumerability...152 // §2. Register Machines...157 // §3. The Halting Problem for Register Machines...163 // §4. The Undecidability of First-Order Logic...167 // §5. Trahtenbrot’s Theorem and the Incompleteness of SecondOrder Logic...170 // §6. Theories and Decidability // Self-Referential Statements and Godel’s Incompleteness Theorems // XI Free Models and Logic Programming // §1. Herbrand’s Theorem...189 // §2. Free Models and Universal Horn Formulas...193 // §3. Herbrand Structures...198 // §4. Propositional Logic...200 // §5. Propositional Resolution...207 // §6. First-Order Resolution (without Unification) ...218 // §7. Logic Programming ...226 // XII An Algebraic Characterization of Elementary Equivalence // 243 // §1. Finite and Partial Isomorphisms...244 // §2. Fraisse’s Theorem ...249 // §3. Proof of Fraisse’s Theorem...251 // §4. Ehrenfeucht Games...258 // XIII Lindstrom’s Theorems 261 // §1. Logical Systems...261 // §2. Compact Regular Logical Systems...264 // §3. Lindstrom’s First Theorem...266 // §4. Lindstrom’s Second Theorem...272 // References 277 // Symbol Index 280 // Subject Index 283