About the Book
This textbook gives a complete and modern introduction to mathematical logic. The author uses contemporary notation, conventions, and perspectives throughout, and emphasizes interactions with the rest of mathematics. In addition to covering the basic concepts of mathematical logic and the fundamental material on completeness, compactness, and incompleteness, it devotes significant space to thorough introductions to the pillars of the modern subject: model theory, set theory, and computability. Requiring only a modest background of undergraduate mathematics, the text can be readily adapted for a variety of one- or two-semester courses at the upper-undergraduate or beginning-graduate level. Numerous examples reinforce the key ideas and illustrate their applications, and a wealth of classroom-tested exercises serve to consolidate readers' understanding. Comprehensive and engaging, this book offers a fresh approach to this enduringly fascinating and important subject.
Table of Contents:
Preface; Acknowledgements; 1. Introduction; 2. Induction and recursion; 3. Propositional logic; 4. First-order logic: languages and structures; 5. Relationships between structures; 6. Implication and compactness; 7. Model theory; 8. Axiomatic set theory; 9. Ordinals, cardinals, and choice; 10. Set-theoretic methods in model theory; 11. Computable sets and functions; 12. Logic, computation, and incompleteness; Appendix: mathematical background; References; Index.
About the Author :
Joseph Mileti is Associate Professor in the Department of Mathematics and Statistics at Grinnell College, USA. He won the 2004 Sacks Prize from the Association of Symbolic Logic for the best doctoral dissertation in mathematical logic (worldwide) and leading experts have celebrated his original courses and materials taught at many universities.
Review :
'This is an excellent introduction to mathematical logic for upper-year undergraduate students with some background in theoretical mathematics. The many wonderful examples relate mathematical logic to other areas of mathematics. Beyond the usual setup and culmination with completeness and incompleteness, Mileti gives a strong introduction to set theory, model theory, and computability. Instructors have the flexibility to suit the examples and direction to their students' backgrounds, and curious students will have an excellent reference for topics not covered in their course. Mileti's gentle prose guides the reader from topic to topic, explaining what we are doing, and why.' Barbara F. Csima, University of Waterloo
'This book is a modern introduction to mathematical logic for a new generation of students. Filled to the brim with examples to motivate students and better explain the myriad topics covered, it is more detailed and complete than almost any text on the subject. Yet, amazingly, it is still quite concise and easy to read. Without a doubt, this is bound to be a new classic.' Damir Dzhafarov, University of Connecticut
'Building on its author's substantial experience teaching courses in the field, this book provides a thorough and wide-ranging introduction to mathematical logic. The careful choice of topics and high level of detail, particularly in early sections covering material that is often challenging even for mathematically well-prepared students, will make it suitable both for one-course introductions to the field and for syllabuses aimed at preparing students for graduate-level courses in subfields such as set theory, model theory, computability theory, and proof theory.' Denis Hirschfeldt, University of Chicago
'Mileti's Modern Mathematical Logic is an absorbing, comprehensive, and well-organized textbook. It contains clear-cut and detailed descriptions of the basic ideas and mathematical arguments, a good number of helpful examples and stimulating exercises, and lucid explanations of the modern view of mathematical logic. It is highly recommended for beginners.' Makoto Kikuchi, Kobe University
'This is a great book. I've used drafts to teach mathematical logic many times in the past and students love it. They find that it is very complete, that it doesn't miss any detail, and that it is a great complement to my informal style of teaching.' Antonio Montalbán, University of California, Berkeley
'I enjoyed using early versions of this text. For mathematics students, it was much better than any of the standard texts. It begins with a general mathematical approach to induction, recursion, and generation to develop both syntax and semantics. It then offers a variety of tailored routes through wide swaths of logic using real mathematical examples. I highly recommend it.' Richard A. Shore, Cornell University
'This book gives a great introduction to mathematical logic for an advanced undergraduate or early graduate-level course. From propositional logic and completeness, through the basics of set theory and model theory, to the incompleteness theorem, the author skillfully presents notions and proofs without shying away from technical details. The exercises after each chapter are at the appropriate level, leading students into more in-depth explorations of various directions related to the topic at hand.' Mariya Soskova, University of Wisconsin-Madison
'This is the logic book I would have written if I could write as clearly as Mileti. At last, a complete introduction to the subject from a modern mathematical perspective that proves the incompleteness theorems while also providing a gentle introduction to model theory, set theory, and computability theory along the way.' Henry Towsner, University of Pennsylvania
Mileti has written a careful, rigorous treatment of a broad range of topics in mathematical logic. This is a valuable addition to the literature. Daniel J. Velleman, Amherst College
'This book is a solid text for the first course on mathematical logic. It contains comprehensive and detailed material on the subject. Although the number of pages of the book is relatively large, any instructor who wants to adopt it as a text should not feel intimidated: the book is organized in such a way that an instructor can choose to emphasize a particular direction, such as set theory, model theory, computability theory, etc., for a one-semester course, or all of them for a one-year course, or anything in between. The proofs of the theorems in the book are gentle and often accompanied with rather illuminating modern examples. There is also a sufficient number of exercises for all sections with various levels of difficulty. Overall, this is an excellent textbook.' Renling Jin, MathSciNet
'This is a new, quite sophisticated, introduction to mathematical logic at the upper undergraduate/beginning graduate level. It is nicely written, adaptable to both semester and year-long courses, and organizationally versatile, and it contains more applications to other branches of mathematics than is typical in books at this level.' Mark Hunacek, The Mathematical Gazette
