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Mathematical Logic Third Edition 2021 [Kietas viršelis]

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  • Formatas: Hardback, 304 pages, aukštis x plotis: 235x155 mm, weight: 698 g, 17 Illustrations, black and white; IX, 304 p. 17 illus., 1 Hardback
  • Serija: Graduate Texts in Mathematics 291
  • Išleidimo metai: 29-May-2021
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030738388
  • ISBN-13: 9783030738389
Kitos knygos pagal šią temą:
Mathematical Logic Third Edition 2021Didesnis paveikslėlis
  • Formatas: Hardback, 304 pages, aukštis x plotis: 235x155 mm, weight: 698 g, 17 Illustrations, black and white; IX, 304 p. 17 illus., 1 Hardback
  • Serija: Graduate Texts in Mathematics 291
  • Išleidimo metai: 29-May-2021
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030738388
  • ISBN-13: 9783030738389
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030738389.html
Raktažodžiai:
This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.





The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.





Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Recenzijos

This newest edition has been reclassified, fittingly, as a graduate text, and it is admirably suited to that role. Those who are already well-versed in logic will find this text to be a valuable reference and a strong resource for teaching at the graduate level, while those who are new to the field will come to know not only how mathematical logic is studied but also, perhaps more importantly, why. (Stephen Walk, MAA Reviews, January 6, 2023)

Part A
I Introduction
3(8)
I.1 An Example from Group Theory
4(1)
I.2 An Example from the Theory of Equivalence Relations
5(1)
I.3 A Preliminary Analysis
6(2)
I.4 Preview
8(3)
II Syntax Of First-Order Languages
11(14)
II.1 Alphabets
11(2)
II.2 The Alphabet of a First-Order Language
13(1)
II.3 Terms and Formulas in First-Order Languages
14(4)
II.4 Induction in the Calculi of Terms and of Formulas
18(5)
II.5 Free Variables and Sentences
23(2)
III Semantics Of First-Order Languages
25(30)
III.1 Structures and Interpretations
26(2)
III.2 Standardization of Connectives
28(2)
III.3 The Satisfaction Relation
30(1)
III.4 The Consequence Relation
31(6)
III.5 Two Lemmas on the Satisfaction Relation
37(4)
III.6 Some Simple Formalizations
41(4)
III.7 Some Remarks on Formalizability
45(4)
III.8 Substitution
49(6)
IV A Sequent Calculus
55(16)
IV.1 Sequent Rules
56(2)
IV.2 Structural Rules and Connective Rules
58(1)
IV.3 Derivable Connective Rules
59(2)
IV.4 Quantifier and Equality Rules
61(2)
IV.5 Further Derivable Rules
63(2)
IV.6 Summary and Example
65(2)
IV.7 Consistency
67(4)
V The Completeness Theorem
71(12)
V.1 Henkin's Theorem
71(4)
V.2 Satisfiability of Consistent Sets of Formulas (the Countable Case)
75(3)
V.3 Satisfiability of Consistent Sets of Formulas (the General Case)
78(3)
V.4 The Completeness Theorem
81(2)
VI The Lowenheim-Skolem And The Compactness Theorem
83(12)
VI.1 The Lowenheim-Skolem Theorem
83(1)
VI.2 The Compactness Theorem
84(2)
VI.3 Elementary Classes
86(4)
VI.4 Elementarily Equivalent Structures
90(5)
VII The Scope Of First-Order Logic
95(16)
VII.1 The Notion of Formal Proof
96(2)
VII.2 Mathematics Within the Framework of First-Order Logic
98(5)
VII.3 The Zermelo-Fraenkel Axioms for Set Theory
103(3)
VII.4 Set Theory as a Basis for Mathematics
106(5)
VIII Syntactic Interpretations And Normal Forms
111(22)
VIII.1 Term-Reduced Formulas and Relational Symbol Sets
111(3)
VIII.2 Syntactic Interpretations
114(6)
VIII.3 Extensions by Definitions
120(4)
VIII.4 Normal Forms
124(9)
Part B
IX Extensions Of First-Order Logic
133(15)
IX.1 Second-Order Logic
133(5)
IX.2 The System
138(5)
IX.3 The System Jzfy
143(5)
X Computability And Its Limitations
147
X.1 Decidability and Enumerability
148(4)
X.2 Register Machines
152(6)
X.3 The Halting Problem for Register Machines
158(5)
X.4 The Undecidability of First-Order Logic
163(2)
X.5 Trakhtenbrot's Theorem and the Incompleteness of Second-Order Logic
165(3)
X.6 Theories and Decidability
168(8)
X.7 Self-Referential Statements and GodePs Incompleteness Theorems
176(6)
X.8 Decidability of Presburger Arithmetic
182(6)
X.9 Decidability of Weak Monadic Successor Arithmetic
188(17)
XI Free Models And Logic Programming
205(52)
XI.1 Herbrand's Theorem
205(4)
XI.2 Free Models and Universal Horn Formulas
209(4)
XI.3 Herbrand Structures
213(3)
XI.4 Propositional Logic
216(6)
XI.5 Propositional Resolution
222(11)
XI.6 First-Order Resolution (without Unification)
233(9)
XI.7 Logic Programming
242(15)
XII An Algebraic Characterization Of Elementary Equivalence
257(16)
XII.1 Finite and Partial Isomorphisms
258(5)
XII.2 FraTsse's Theorem
263(2)
XII.3 Proof of Frai'sse's Theorem
265(6)
XII.4 Ehrenfeucht Games
271(2)
XIII Lindstrom's Theorems
273(18)
XIII.1 Logical Systems
273(3)
XIII.2 Compact Regular Logical Systems
276(2)
XIII.3 Lindstrom's First Theorem
278(7)
XIII.4 Lindstrom's Second Theorem
285(6)
References 291(2)
List of Symbols 293(4)
Subject Index 297
Heinz-Dieter Ebbinghaus is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His work spans fields in logic, such as model theory and set theory, and includes historical aspects.Jörg Flum is Professor Emeritus at the Mathematical Institute of the University of Freiburg. His research interests include mathematical logic, finite model theory, and parameterized complexity theory.





Wolfgang Thomas is Professor Emeritus at the Computer Science Department of RWTH Aachen University. His research interests focus on logic in computer science, in particular logical aspects of automata theory.