A Friendly Introduction to Mathematical Logic 2nd Edition by Christopher C. Leary (PDF)

Description

At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary’s user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition’s treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel’s First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This may well be the finest introduction to mathematical logic that you have never heard of. The current edition is published in-house at SUNY Geneseo, and has therefore not received the same promotional advantages of books from the larger publishers. This is one reason why I am placing this review on Amazon; in my opinion, this extraordinary book deserves to be widely used . I recommend this book in the strongest possible terms to two groups: (1) mathematics professors searching for a text for the first undergraduate course in mathematical logic, and (2) mathematically mature readers who want to study mathematical logic alone.I have taught mathematical logic to undergraduates many times over the years. As primary course texts, I have used three well-known books: (1) Enderton; (2) Ebbinghaus, Flum and Thomas; and (3) Hodel. I admire each of these books, and class usage has clearly revealed that each has its own distinct strengths.. However, after working through the first six chapters of Leary and Kristiansen’s ([L-K]) book rather carefully, including writing up solutions to all the exercises, I have decided that this will become my primary course text the next time I teach mathematical logic. What follows are some of the characteristics of the book that led me to that decision.1) The User-Friendly Tone of Presentation: Reading this text is somewhat like receiving personal tutoring from your elder sibling who has received a Ph.D. in mathematical logic. In places when it is appropriate, the prose is quite informal, explaining complicated concepts in intuitive terms, providing historical details to help motivate, and helping the student navigate through lengthy passages of rather detailed proofs and constructions with frequent words of explanation and encouragement. The authors’ enthusiasm for the subject is apparent on nearly every page. As [L-K] state in the Preface, their primary goal is to permit students to study the Completeness and Soundness Theorems, The Compactness Theorem, and Godel’s First and Second Incompleteness Theorems in a traditional, one-semester course. Some might believe that this is an impossibly ambitious goal (I did, at first). But after careful study of the book, I am convinced the authors have succeeded. In order to achieve these goals, the authors have devoted less space than other books do to certain foundational topics (even the customary first chapter on propositional logic is missing). After years of experience, I now agree with the wisdom (and necessity) of this decision; other instructors may disagree, of course. My goals in a first mathematical logic course have evolved over the years to exposing the students to a few of the most compelling results so that, just perhaps, they will be motivated to continue their study of logic long after the course ends. Will students be experts at using the many versions of the General Recursion Theorem after reading this book? Certainly not. But they will be well prepared to continue their studies in a graduate text such as HInman or Bell and Machover, where these topics will be fully developed and, perhaps, more fully appreciated.2) The Clarity of the Writing: The first author states in the Preface, “One of the tasks that I set for myself as I wrote this book was to be mindful of the audience.” The result is a book that a motivated undergraduate mathematics major could use for independent study (a rarity in this difficult subject). The perfect example to illustrate the point is the proof of the Completeness Theorem, which begins on page 74 and doesn’t conclude until page 86. As many authors do, [L-K] present Leon Henkin’s beautiful proof of this result. However, their presentation is quite unique. A global outline of the proof is presented first to help the student fix in his mind the path of the entire journey. The proof is then presented in small, incremental steps, with each step explained in clear terms before the formal mathematics occurs. One can actually read all twelve pages of this proof in one sitting, follow and understand both the details and the IDEA of the proof, and not feel overwhelmed by the effort. This is a masterful presentation. In his popular book on the Theory of Computation, Michael Sipser introduced the “Idea of Proof” passages that precede actual proofs. Leary and Kristiansen take this innovative idea to another level entirely, inserting clear discussions of the ideas within each stage of the formal proofs, just at the points where they are most needed.The book also contains 54 indented passages, labeled “Chaff”, which are meant to clarify, amplify, foreshadow, and/or explain especially difficult points. The authors state that, like chaff, these passages could blow away in the wind and all the requisite mathematical details would still be there; however, I predict with confidence that students will find the Chaff sections the most helpful and informative passages in the text. As I read many of these passages, I found myself thinking, “That is exactly what I would have said in my lecture to help clarify the point.”3) The Exercises: It is apparent that serious thought (and some restraint) went into the selection of the exercises, which are integrated perfectly within the text. The first seven chapters contain a total of 234 exercises, with an average of six questions per set. The serious student therefore could, and should, attempt all of them without undue stress. In many sections, selected exercises complete missing steps within a proof in the text, thus inspiring a careful reading of the proof. The level of difficulty is adequate to challenge students without overwhelming them. Answers to many (not all) of the exercises are provided in the back of the book; this will be extremely helpful to those who study alone.The addition of the second author for the second edition resulted in the inclusion of much new material on the theory of computation. Accordingly, the text has been organized so that instructors can choose either one of two approaches to Godel’s First Incompleteness Theorem without loss of continuity.Prospective readers always want to know about prerequisites. In my judgment, this book is accessible to any student of mathematics who has had (1) a transition course introducing proofs and propositional logic, followed by (2) at least one solid, proof-based course in linear algebra, abstract algebra, or elementary analysis. It is more an issue of maturity than specific material prerequisites. However, many of the structures that are studied are drawn from abstract algebra.Not all mathematicians study logic and foundations as part of their education, so the target audience for any such text is relatively small. SUNY-Geneseo should be commended for making this exceptional textbook available to those who want to study the foundations of mathematics.

⭐On the whole it is a friendly introduction.Occasionally I could wish that terms were defined more explicitly early on. The approach for say “non-logical axioms” seem to be let the context somewhat supply the definition. And then define them precisely when a detailed discussion of them is reached. Even a limited early temporary definition would have given me at least the illusion of greater understanding sooner.But it is generally a friendly introduction on a math upperclassman or beginning graduate student level.And Amazon delivered it (free shipping with Prime) on Tuesday after I ordered over the weekend (Sunday I think.).

⭐This book was a great way to learn about Math Logic. I do admit that I had some experience coming from a Discrete Math & Theory of Computation background, but this book was extremely accessible. None of this garbage like “A proof is a 7 tuple consisting of …” that makes no sense. The authors are very witty, and they make the material easy to learn. Not to say that it isn’t hard – there are some sections that you may have to read a few times to understand, but the material is all there 🙂

⭐I am not sure what is exactly ‘friendly’ about this book. The title is QUITE the misrepresentation of the content of the book. The authors, by no means, are trying to simplify things or help you study the material on your own as some reviewers here stated. The examples are hardly illustrative and they are stated in an almost sarcastic way that makes one believe that the author is subconsciously deriving some sort of satisfaction from providing an example that does not serve the purpose of being an illustrative example !!!The writer clearly lacks the ability to put himself in the shoes of the student and, from that perspective, decide how to lead the student to where he wants to lead him. THAT is the skill that distinguishes a good teacher from one of a lower caliber. And the writer of this book most certainly belongs to the latter group.I am very upset that I paid that much money for a totally useless book like that. Putting ‘friendly’ in the title is extreme customer deception. I wish I could return this book.

⭐It’s friendly enough if you have an undergraduate background in advanced mathematics.

⭐Not all that friendly. It has a fault of using the exercises in the text and no answers for them.

⭐The impossible goal of this text is to start from scratch and then cover both incompleteness theorems in a single semester, and under his presentation it would almost be manageable. This is by far the best written text on predicate calculus I have read. Kaye and Goldrei can’t really compare, as they contain less material and what they do cover isn’t done quite as well. Enderton on the other hand covers more than Leary, but is much more dense and would not serve as well as an introduction.The main drawback of the book is how much effort the author put into making it fit into a single semester. There is a lot of fascinating material that could have been covered in greater depth than is done. It is worth noting that he almost completely skips over propositional calculus, so if you find yourself struggling at the beginning of the book you may want to read up on that subject in another text (the first half of Goldrei would do nicely). Also the section on the second incompleteness theorem is extremely rushed; some of the properties of peano arithmetic used for the proof are not proven.Still, it’s better than the other options I’ve seen. You would think with all the mediocre mathematics texts Dover picks up they would have found this gem.Beyond knowing the rudiments of naïve set theory, prerequisites should not be much of a concern.

⭐I have been trying to teach myself metalogic and mathematical logic from a standing start. This book isn’t for beginners like me, but it is nicely presented so I shall keep it as a reference book for comparison with others on the same subject, of which I would recommend Geoffrey Hunter’s Metalogic. One slight irritation is a joke at the beginning of the text about the immaculate conception. The author clearly doesn’t know what the immaculate conception is so he’d be better off trying to raise a laugh by sticking to things he knows about.

⭐Ich habe bisher die ersten 3 Kapitel durchgearbeitet, ‘Structures and Languages’, ‘Deductions’ und ‘Completeness and Compactness’. Bisher komme ich mit dem Buch sehr gut zurecht. Hintergrund: ich möchte mich in die Lage versetzen die Gödel’schen Unvollständigkeitssätze und ihre Beweise zu verstehen.Wer sich für Mathematische Logik interessiert und ein wenig Erfahrung im Durcharbeiten mathematischer Texte hat, wird für sein Durchhaltevermögen belohnt. Letzteres ist aber auf jeden Fall gefragt, denn obwohl ‘friendly’ im Titel steht, ist das Durcharbeiten mit einigem Aufwand verbunden, was allerdings der Materie geschuldet ist und nicht dem Autor, der den Stoff einfach und mit großer Klarheit präsentiert.I purchased this book on behalf of a friend, so, I’m afraid I couldn’t possibly make a personal comment. However, I’m told it served its purpose.

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