A concise introduction to mathematical logic

Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics. The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Gdels Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the authors website.

[1]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[2]  G. L. Collected Papers , 1912, Nature.

[3]  Stephen Read,et al.  FROM MATHEMATICS TO PHILOSOPHY , 1974 .

[4]  R. McKenzie,et al.  A CHARACTERIZATION OF FINITELY DECIDABLE CONGRUENCE MODULAR VARIETIES , 1994 .

[5]  A. Pillay Models of Peano Arithmetic , 1981 .

[6]  W. Luxemburg Non-Standard Analysis , 1977 .

[7]  M. E. Szabo,et al.  The collected papers of Gerhard Gentzen , 1969 .

[8]  Elliott Mendelson,et al.  Introduction to Mathematical Logic , 1979 .

[9]  G. Birkhoff,et al.  On the Structure of Abstract Algebras , 1935 .

[10]  Wolfgang Rautenberg,et al.  Klassische und nichtklassische Aussagenlogik , 1979 .

[11]  Alonzo Church,et al.  A note on the Entscheidungsproblem , 1936, Journal of Symbolic Logic.

[12]  Konstantin N. Ignatiev,et al.  On strong provability predicates and the associated modal logics , 1993, Journal of Symbolic Logic.

[13]  Hans Hermes,et al.  Introduction to mathematical logic , 1973, Universitext.

[14]  Olgierd Wojtasiewicz,et al.  Elements of mathematical logic , 1964 .

[15]  Gottlob Frege,et al.  Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens , 1879 .

[16]  A. Robinson I. Introduction , 1991 .

[17]  M. H. Lob,et al.  Solution of a Problem of Leon Henkin , 1955, J. Symb. Log..

[19]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[20]  David Marker,et al.  Introduction to Model Theory , 2000 .

[21]  B. Poizat A Course in Model Theory , 2000 .

[22]  K. Gödel Die Vollständigkeit der Axiome des logischen Funktionenkalküls , 1930 .

[23]  A. S. Troelstra,et al.  Ω-Bibliography of Mathematical Logic , 1987 .

[24]  Alfred Horn,et al.  On sentences which are true of direct unions of algebras , 1951, Journal of Symbolic Logic.

[25]  A. Macintyre,et al.  Gödel's diagonalization technique and related properties of theories , 1973 .

[26]  Wolfram Pohlers Proof Theory: An Introduction , 1990 .

[27]  Wilfried Sieg Herbrand analyses , 1991, Arch. Math. Log..

[28]  J. Lloyd Foundations of Logic Programming , 1984, Symbolic Computation.

[29]  Lev D. Beklemishev,et al.  ON THE CLASSIFICATION OF PROPOSITIONAL PROVABILITY LOGICS , 1990 .

[30]  D. C. Cooper,et al.  Theory of Recursive Functions and Effective Computability , 1969, The Mathematical Gazette.

[31]  Azriel Levy Basic set theory , 1979 .

[32]  Hilary Putnam,et al.  Philosophy of mathematics : selected readings , 1984 .

[33]  Kenneth Kunen,et al.  Set Theory: An Introduction to Independence Proofs , 2010 .

[34]  K. Gödel Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I , 1931 .

[35]  Dirk van Dalen,et al.  Logic and structure , 1980 .

[36]  Shashi M. Srivastava,et al.  A Course on Mathematical Logic , 2008, Universitext.

[37]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[38]  Lev D. Beklemishev Iterated Local Reflection Versus Iterated Consistency , 1995, Ann. Pure Appl. Log..

[39]  J. Heijenoort From Frege To Gödel , 1967 .

[40]  Per Lindström,et al.  On Extensions of Elementary Logic , 2008 .

[41]  S. Buss Handbook of proof theory , 1998 .

[42]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[43]  Leon Henkin,et al.  The completeness of the first-order functional calculus , 1949, Journal of Symbolic Logic.

[44]  Hao Wang,et al.  A Logical Journey: From Gödel to Philosophy , 1996 .

[45]  Walter Felscher Lectures on mathematical logic , 2000 .

[46]  A. Tarski A Decision Method for Elementary Algebra and Geometry , 2023 .

[47]  Ivan Korec Model-interpretability into trees and applications , 1975, Arch. Math. Log..

[48]  I. S. Gradshtein,et al.  THE ELEMENTS OF MATHEMATICAL LOGIC , 1963 .

[49]  L. M.-T. Grundzüge der theoretischen Logik , 1929, Nature.

[50]  Wolfgang Rautenberg,et al.  Finite Replacement and Finite Hilbert-Style Axiomatizability , 1992, Math. Log. Q..

[51]  Melvin Fitting Incompleteness in the Land of Sets , 2007 .

[52]  A. Selman Completeness of calculii for axiomatically defined classes of algebras , 1972 .

[53]  Peter Smith,et al.  An Introduction to Gödel's Theorems , 2007 .

[54]  J. D. Monk,et al.  Mathematical Logic , 1976 .

[55]  J. Barkley Rosser,et al.  Extensions of some theorems of Gödel and Church , 1936, Journal of Symbolic Logic.

[56]  John W. Dawson,et al.  Logical dilemmas - the life and work of Kurt Gödel , 1996 .

[57]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[58]  A. Turing On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[59]  Gerald E. Sacks,et al.  Saturated Model Theory , 1972 .

[60]  D.H.J. de Jongh,et al.  The logic of the provability , 1998 .

[61]  D. Gabbay Decidability results in non-classical logic. III , 1971 .

[62]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[63]  Albert Visser,et al.  An Overview of Interpretability Logic , 1997, Advances in Modal Logic.

[64]  Anand Pillay,et al.  Simple Theories , 1997, Ann. Pure Appl. Log..

[65]  C. Ryll-Nardzewski The role of the axiom of induction in elementary arithmetic , 1952 .

[66]  Ralph McKenzie,et al.  Structure of decidable locally finite varieties , 1989, Progress in mathematics.

[67]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[68]  G. Boolos,et al.  Self-Reference and Modal Logic , 1985 .

[69]  Alfred Tarski,et al.  Der Wahrheitsbegriff in den formalisierten Sprachen , 1935 .

[70]  Stewart Shapiro,et al.  The Oxford Handbook of Philosophy of Mathematics and Logic , 2005, Oxford handbooks in philosophy.

[71]  Kees Doets,et al.  From logic to logic programming , 1994, Foundations of computing series.

[72]  A. Wilkie Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function , 1996 .

[73]  Giovanni Sambin,et al.  An effective fixed-point theorem in intuitionistic diagonalizable algebras , 1976 .

[74]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[75]  John L. Bell,et al.  A course in mathematical logic , 1977 .

[76]  E. Groves A Dissertation ON , 1928 .

[77]  Wilhelm Ackermann,et al.  Die Widerspruchsfreiheit der allgemeinen Mengenlehre , 1937 .

[78]  Peter Aczel etc HANDBOOK OF MATHEMATICAL LOGIC , 1999 .

[79]  Lev D. Beklemishev,et al.  Bimodal logics for extensions of arithmetical theories , 1996, Journal of Symbolic Logic.

[80]  Jacques Herbrand Recherches sur la théorie de la démonstration , 1930 .

[81]  Richard Statman,et al.  Logic for computer scientists , 1989 .

[82]  S. Feferman Arithmetization of metamathematics in a general setting , 1959 .

[83]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[84]  M. Kracht Tools and Techniques in Modal Logic , 1999 .

[85]  W. Hodges CLASSIFICATION THEORY AND THE NUMBER OF NON‐ISOMORPHIC MODELS , 1980 .

[86]  Petr Hájek,et al.  Metamathematics of First-Order Arithmetic , 1993, Perspectives in mathematical logic.

[87]  Jeff B. Paris,et al.  On the scheme of induction for bounded arithmetic formulas , 1987, Ann. Pure Appl. Log..

[88]  Harvey M. Friedman,et al.  Elementary Descent Recursion and Proof Theory , 1995, Ann. Pure Appl. Log..

[89]  D. Marker Model theory : an introduction , 2002 .

[90]  José Sanmartín Esplugues Introduction to model theory and to the metamathematics of algebra , 1971 .

[91]  H. Keisler Logic with the quantifier “there exist uncountably many” , 1970 .

[92]  R. Smullyan Theory of formal systems , 1962 .

[93]  Gregory H. Moore The emergence of first-order logic , 1988 .

[94]  R. Tennant Algebra , 1941, Nature.

[95]  P. Rothmaler Introduction to Model Theory , 2000 .

[96]  Wanda Szmielew Elementary properties of Abelian groups , 1955 .

[97]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .