# Mathematical Logic

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

### 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

### 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, ...

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/