Mathematical Logic (Dover Books on Mathematics) by Stephen Cole Kleene (PDF)

Description

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text. It begins with an elementary but thorough overview of mathematical logic of first order. The treatment extends beyond a single method of formulating logic to offer instruction in a variety of techniques: model theory (truth tables), Hilbert-type proof theory, and proof theory handled through derived rules.The second part supplements the previously discussed material and introduces some of the newer ideas and the more profound results of twentieth-century logical research. Subsequent chapters explore the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel’s completeness theorem, Gentzen’s theorem, Skolem’s paradox and nonstandard models of arithmetic, and other theorems. The author, Stephen Cole Kleene, was Cyrus C. MacDuffee Professor of Mathematics at the University of Wisconsin, Madison. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index.

User’s Reviews

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

⭐This book was written by one of the great American mathematical minds of this century. I’ve read it cover to cover and it happens to be my favorite logic book for its scope, depth, and clarity. Kleene uses a combined model-theoretic and proof-theoretic approach, and derives many interesting results relating the two (he also gives mention to special axioms for Intuitionistic logic). Although his focus in the first part of the book is on a more or less mathematical treatment of standard first-order predicate logic (augmented later by functions and equality), he also spends considerable time discussing the ways in which formal logic can and should be used to analyze “ordinary language” statements and arguments. After setting the groundwork, he moves onto subjects such as set theory, formal axiomatic theories, turing machines and recursiveness, Godel’s incompleteness theorem, Godel’s completeness theorem, and just about every interesting subject relating to logic in the first half of the twentieth century.For the mathematically inclined self-teacher, Kleene’s exposition should not be difficult at all, in fact I found it remarkably clear compared to other mathematical treatments of the subject (which are necessary if one wants to understand the deeper results). I suppose less mathematically inclined readers could try Irving Copi’s “Symbolic Logic” as a start, although even that requires some mathematical proficiency, and since it doesn’t cover many of the things you will want to know about, you’ll end up coming back to a book like Kleene’s anyway. So to summarize, if you want to learn the hard stuff (from the first half of the twentieth century–which includes just about everything the layman/philosopher wants to know), there is no better or easier way.

⭐I bought this book based on the short altough extremely positive review in the bibliography section of Hofstadter’s GEB. And it has proven to be one of the my best acquisitions. The books is wonderfully written, very detailed indeed, so it makes it easy to follow the proofs. It comes along with exercises at the end of every paragraph (chapters are divided into paragraphs) so you can process and reinforce what has been learned.I have worked with plenty of logic books (among them: Suppes’, Gamut’s, Machover’s, etc) but I am loyal to Kleene’s.What it attracks me the most is that it not only contains detailed and rigorous proofs of the most important theorems of logic, but that it also comes with philosophical considerations of one of the best logician’s of the 20th century. There are other parts too where Kleene includes a piece of history, for instance, about the Löwenheim-Skolem theorem: delightful.Buy it ’cause you won’t regret it.

⭐My problem with Kleene’s Mathematical Logic is simply that its layout is terrible. There is hardly any white space in the text, so the entire book reads like a giant run-on sentence. This makes it difficult to find information that you want when you want to refer to a previous topic. There is also no glossary (defined terms are italicized in otherwise normal sentences, and hardly stand out at all. I generally highlighted them, since that was my only hope for referring to them later, short of typing my own glossary). There is also no answer key, which in my opinion is an extremely important element of any textbook.That being said, all of the information that you could want is there. Simple concepts are built up appropriately before more abstract ones are given. Proofs of all important theorems are provided. Examples and exercises are abundant. In order to motivate you, the author simply insults your intelligence every once and a while, and you are thus determined to prove him wrong (I actually did enjoy this aspect of Kleene’s style).In summary – If you are determined, you can teach yourself mathematical logic from this book. However, you can make this undertaking much easier on yourself by getting a more readable textbook.

⭐I was looking for a fairly rigorous introduction to mathematical logic that treated foundational issues (Godel’s theorem, decidabilty, etc.). Stephen Kleene is well-qualified to write one, having done some work in the 1930s related to the lamda calculus. The book is comprehensive and more sophisticated and detailed than I needed, but you can’t blame the book for that. I’ve given it a good/fair rating because the exposition and organization could have been better. I would recommend W.V.O. Quine’s Introduction to Mathematical Logic, not as an alternative exactly, because it’s much more limited and doesn’t address foundational issues, but because of its remarkable clarity and concision.

⭐Everything you always wanted to know!

⭐Simply started into it since receiving it and I find it fascinating from the get go, and thoroughly so. If the rest of the book carries for in that spirit I’ll be overjoyed.

⭐I like this kind of stuff. It’s so ‘touchy feely’ andlike a warm pair of slippers on a cold country night.WHAT THE HELL AM I TALKING ABOUT?THIS STUFF IS HARD.

⭐Want to really understand mathematical logic? This is what you need.

⭐Excellent

⭐Estoy muy satisfecho con la compra de este libro. El autor explica muy bien los conceptos más básicos de lógica (conectores lógicos, tablas de verdad, tautologías, etc.) pasando por la lógica de primer orden, hasta llegar a su aplicación en teoría de conjuntos o en la exposición de los teoremas de Godel.Creo que es un libro bastante completo que empieza desde lo más básico. Un aspecto negativo a resaltar es que el libro contiene páginas sobrecargadas de texto y leerlo puede resultar incómodo.Lo recomiendo.A brilliant read that gives a very fundamental perspective of the subject from the eyes of an expert. I particularly love the level of formalism used in the book. Its high enough that the subject is not rendered imprecise as any slightly less formal treatment risks doing. At the same time, the formalism is not too strict that it makes the text unapproachable and daunting.

⭐testo didatticamente efficace,ma introduttivo alla materia non trattata in modo completoI would not add much by saying that “Introduction to MetaMathematics” (IM) remains a masterpiece, even though the style is a bit oldish…On the the other hand, “Mathematical Logic” (ML) brings a definite plus, but is by no means a replacement, rather a necessary complement.As I planned to study both, the problem posed was the order in which one should approach those books : Historically ? By increasing or decreasing difficulty ? In parallel, in order to see how Kleene’s ideas — and the field — have evolved between 1952 and 1966, and subject by subject ?I chose the third an most difficult path… And the journey was a thrill !Here is how I planned this strange exploration : IM, ch. 1 to 7 ; ML, ch. 1 to 4 ; IM, ch. 8 ; IM, Part III ; ML, ch. 5 : IM, ch. 14 ; ML, ch. 6 ; IM, ch. 15.ML is certainly less difficult but contains a fair amount of footnotes linking it to IM, i.e. studying IM is simply inevitable and enjoyable, even though some parts are really tough and must be “examined in a cursory manner”, as suggested by Kleene, e.g. ch. 14 & 15.IM, part III, is a thorough treatment of recursive functions, the best in my opinion and is not part of ML.All in all, the two together rank very high in logic books, perhaps highest.This book now stands in my list of outstanding books on logic :1. A. Tarski’s “Introduction to Logic”, a jewel, followed by P. Smith’s superb entry-point “An introduction to Formal logic” and the lovely “Logic, a very short introduction” by Graham Priest2. D. Goldrei’s “Propositional and Predicate calculus”3. Wilfrid Hodges’ “Logic”, followed by Smullyan’s “First-order logic”.4. P. Smith’s “An introduction to Gödel’s theorems”.5. Kleene’s “Introduction to metamathematics” & “Mathematical Logic”.6. G. Priest’s ” Introduction to non-classical logic”.Hence forgetting altogether Van Dalen’s indigestible “Logic & Stucture” as well asthe even more indigestible Enderton, Mendelson & al…

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