Mathematical Logic (Dover Books on Mathematics)
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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.
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.
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.
Top international reviews
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.
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...