Mathematical Logic

I 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 Priest 2. 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 as the even more indigestible Enderton, Mendelson & al...

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.

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.

This is one of the best books on mathematical logic ever written. The author himself contributed to the foundation of computability theory, so the book is delves in that direction. The dramatis personae are a variegated exposition of formal methods beginning with truth tables, Hilbert axiomatics, Gentzen systems, logical number theory, a brief chapter on Turing Machines, Undecidability results, and a thorough exposition of Godel's Theorem. It's salient point is conceptual clarity of the notions involved througout the mastery of formal notions.

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.