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Mathematical Logic

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Springer, Jun 10, 1994 - Mathematics - 289 pages
This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic.
  

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User Review  - Joecolelife - Goodreads

The formal mathematics is organized and presented so clearly and precisely that I felt I was admiring a fine crystal structure. The notation used may seem excessive to some, but it actually is the ... Read full review

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Contents

Introduction
3
1 An Example from Group Theory
4
2 An Example from the Theory of Equivalence Relations
5
3 A Preliminary Analysis
6
4 Preview
8
Syntax of FirstOrder Languages
11
2 The Alphabet of a FirstOrder Language
13
3 Terms and Formulas in FirstOrder Languages
15
4 Set Theory as a Basis for Mathematics
110
Syntactic Interpretations and Normal Forms
115
2 Syntactic Interpretations
118
3 Extensions by Definitions
125
4 Normal Forms
128
PART B
135
Extensions of FirstOrder Logic
137
1 SecondOrder Logic
138

4 Induction in the Calculus of Terms and in the Calculus of Formulas
19
5 Free Variables and Sentences
24
Semantics of FirstOrder 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
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
The Completeness Theorem
75
2 Satisfiability of Consistent Sets of Formulas the Countable Case
79
3 Satisfiability of Consistent Sets of Formulas the General Case
82
4 The Completeness Theorem
85
The LöwenheimSkolem Theorem and the Compactness Theorem
87
2 The Compactness Theorem
88
3 Elementary Classes
91
4 Elementarily Equivalent Structures
94
The Scope of FirstOrder Logic
99
2 Mathematics Within the Framework of FirstOrder Logic
103
3 The ZermeloFraenkel Axioms for Set Theory
107
2 The System ℒw₁w
142
3 The System ℒQ
148
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 FirstOrder Logic
167
5 Trahtenbrots Theorem and the Incompleteness of SecondOrder Logic
170
6 Theories and Decidability
173
7 SelfReferential Statements and Gödels Incompleteness Theorems
181
Free Models and Logic Programming
189
2 Free Models and Universal Horn Formulas
193
3 Herbrand Structures
198
4 Propositional Logic
200
5 Propositional Resolution
207
6 FirstOrder Resolution without Unification
218
7 Logic Programming
226
An Algebraic Characterization of Elementary Equivalence
243
1 Finite and Partial Isomorphisms
244
2 Fraïssés Theorem
249
3 Proof of Fraïssés Theorem
251
p Ehrenfeucht Games
258
Lindströms Theorems
261
2 Compact Regular Logical Systems
264
3 Lindströms First Theorem
266
4 Lindströms Second Theorem
272
References
277
Symbol Index
280
Subject Index
283
Copyright

Other editions - View all

Mathematical logic
Mathematical logic
Heinz-Dieter Ebbinghaus,Jörg Flum,Wolfgang Thomas
Snippet view - 1984
Mathematical Logic
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H.-D. Ebbinghaus,J. Flum,W. Thomas
No preview available - 1994
Mathematical logic
Mathematical logic
Heinz-Dieter Ebbinghaus,Jörg Flum,Wolfgang Thomas
No preview available - 1994

Common terms and phrases

Algebra algorithm alphabet applied arbitrary assignment axiom system binary relation Chapter clauses Compactness Theorem consistent set countable decidable define definition derivable disjunction dom(p domain elementarily equivalent elementary elements enumerable equality-free example Exercise finite subset first-order language first-order logic first-order sentence following holds formal function symbol given Godel H-derivation hence Herbrand structure Horn sentences induction hypothesis infinite logical system logically equivalent Lowenheim-Skolem Theorem mathematical n-ary natural numbers nonempty notion obtain ordered field partial isomorphism prenex normal form procedure proof propositional logic propositional variables prove quantifier quantifier-free R-decidable R-enumerable real numbers rules S-formula S-sentence S-structures 21 S-terms satisfiable second-order second-order logic sequent calculus set of formulas set of sentences set theory strings Suppose symbol set syntactic interpretation system of axioms unary relation symbol uncountable universal Horn formulas

References to this book

From other books

Finite Model Theory
Finite Model Theory
Heinz Dieter Ebbinghaus,Jörg Flum
Limited preview - 1999
Handbook of Formal Languages: Beyond Words
Handbook of Formal Languages: Beyond Words
Grzegorz Rozenberg,Arto Salomaa
No preview available - 1997
All Book Search results »

From Google Scholar

Logic and the Challenge of Computer Science
YURI GUREVICH
Expressiveness of Structured Document Query Languages Based on ...
FRANK NEVEN, JAN VAN DEN BUSSCHE - 2002 - Journal of the ACM
A first-order axiomatization of the theory of finite trees
Rolf Backofen, James Rogers, K Vijay-Shanker - 1995 - Journal of Logic, Language and Information
Monadic second-order logic over pictures and recognizability by ...
Dora Giammarresi, Sebastian Seibert, Antonio Restivo, Wolfgang Thomas
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References from web pages

Structure (mathematical logic) - Wikipedia, the free encyclopedia
Structure (mathematical logic) ..... A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Berlin, New York: Springer-Verlag, ...
en.wikipedia.org/ wiki/ Structure_(mathematical_logic)

Ebbinghaus, Flum, Thomas. Mathematical Logic.
Chapter 1: Introduction--provides motivational text (distinguishing between traditional philosophical logic and mathematical logic) and motivational ...
mathgate.info/ cebrown/ notes/ ebbinghaus.php

JSTOR: Mathematical Logic
It broadly follows the lines that are taken by the majority of mathematical logic text books as regards choice of topics, notation, definitions, ...
links.jstor.org/ sici?sici=0025-5572(198506)2%3A69%3A448%3C147%3AML%3E2.0.CO%3B2-Z

Ebbinghaus H.-D., Flum J., Thomas W. Mathematical logic (ISBN 0 ...
Ebbinghaus H.-D., Flum J., Thomas W. Mathematical logic (ISBN 0-387-90895-1)(Springer, 1984)(L)(T)(113s).djvu. Size 2.1Mb Date Sep 17, 2004 ...
www.eknigu.org/ info/ M_Mathematics/ MA_Algebra/ MAml_Mathematical%20logic/ Ebbinghaus%20H.-D.,%20Flum%20J.,%20Thomas%20W...

Basic Library List-Foundations and Mathematical Logic
Andrews, Peter B. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof New York, NY: Academic Press, 1986. ...
www.maa.org/ BLL/ foundations.htm

CIDEC Library: Ebbinghaus * Mathematical Logic
MATHEMATICAL LOGIC 2nd ed. Uniform Title: Einführung in die mathematisch Logik. English Translated from the German by Ferebee, as ...
cs.ioc.ee/ yik/ lib/ 1/ Ebbinghaus2.html

MATHEMATICAL STRUCTURES RESEARCH
"Axioms for Abstract Model Theory",Annals of Mathematical Logic 7(1974) 221-265. ... Mathematical Logic. Springer-Verlag 1984. Ebbinghaus and Flum. ...
www.mmsysgrp.com/ mathstrc.htm

FINITE MODEL THEORY (Perspectives in Mathematical Logic) By Heinz ...
FINITE MODEL THEORY (Perspectives in Mathematical Logic) By Heinz-Dieter Ebbinghaus and Jorg Flum: 327 pp., DM. 148.–, ISBN 3 540 60149 X (Springer, 1995). ...
journals.cambridge.org/ abstract_S0024609396222416

Abstract Math: Mathematical Reasoning
Mathematical logic (or proof theory) is a branch of mathematics that uses mathematical ... Mathematical logic is quite technical but very powerful; ...
www.abstractmath.org/ MM/ MMMathReasoning.htm

18.511: Mathematical Logic
Description, This course provides an introduction to mathematical logic. ... Text, Mathematical Logic, by Ebbinghaus, Flum, and Thomas (Springer 1994) ...
www-math.mit.edu/ ~rosen/ 18.511/

About the author (1994)

Prof. Jvrg Flum, Abteilung f]r Mathematische Logik, Albert-Ludwigs-Universitdt Freiburg, Germany, http: //logik.mathematik.uni-freiburg.de/personen/Flum.html Prof. Martin Grohe, Institut f]r Informatik, Humboldt-Universitdt zu Berlin, Germany, http: //www.informatik.hu-berlin.de/~grohe/ The authors are very well qualified to write this book. In addition to their strong backgrounds in complexity, algorithms, etc., they have contributed a number of specific key results in parameterized complexity (e.g., http: //epubs.siam.org/sam-bin/dbq/article/42720). Jvrg Flum has coauthored two other Springer monographs: (i) "Mathematical Logic," Undergraduate Texts in Mathematics, 0-387-94258-0, 3rd printing since 1994, over 4000 copies sold, Heinz-Dieter Ebbinghaus, Jvrg Flum, Wolfgang Thomas, http: //www.springer.com/0-387-94258-0. (ii) "Finite Model Theory," Springer Monographs in Mathematics (was in series Perspectives in Mathematical Logic), printed in soft- and hardback, 1995, 2nd ed. in 1999, 2nd corr. print in 2006, Heinz-Dieter Ebbinghaus, Jvrg Flum, 3-540-28787-6, http: //www.springer.com/3-540-28787-6. In addition, Jvrg Flum coauthored the following LNM title: Vol. 769, "Topological Model Theory, 1980, 3-540-09732-5, Jvrg Flum, Martin Ziegler. And he coedited the following LNCS title: Vol. 1683, CSL 1999 conf. proc., Jvrg Flum, Mario Rodriguez-Artalejo, 1999, 3-540-66536-6. Prof. Martin Grohe has authored over 50 articles for refereed theoretical computer science journals and conference proceedings (http: //www.informatik.uni-trier.de/~ley/db/indices/a-tree/g/Grohe: Martin.html) in the areas of logic, complexity, algorithms, etc.

Prof. Dr. Heinz-Dieter Ebbinghaus und Prof. Dr. JArg Flum forschen und lehren am Institut fA1/4r Mathematik der UniversitAt Freiburg, Prof. Dr. Wolfgang Thomas ist Inhaber des Lehrstuhls fA1/4r Informatik 7 (Logik und Theorie diskreter Systeme) der RWTH Aachen.

Bibliographic information

QR code for Mathematical Logic
TitleMathematical Logic
Springer Undergraduate Texts in Mathematics and Technology
Undergraduate Texts in Mathematics, ISSN 0172-6056
AuthorH.-D. Ebbinghaus
ContributorsJ. Flum, Wolfgang Thomas
Edition2, illustrated
PublisherSpringer, 1994
ISBN0387942580, 9780387942582
Length289 pages
Subjects › 

Mathematics / Logic
  
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