Customer Reviews

Introduction to Metamathematics

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This is one of those books that don't get old; although it was first published in 1952, and since then much has been made in Mathematical Logic, Kleene's book has that rare position of a book that influenced the subject on its own (and all the teaching books that came after). And if you are willing to understand Mathematical Logic, and principally the reasons behind most of the definitions, I think that this is the best book to start. As a reference it is perhaps the most cited book in the area. But the reading is pleasant, elegant and well motivated. This book has another kind of appeal, in my opinion - research in Logic split after the 1950's in two distinct areas: one, more mathematical in character, is called Model Theory and is strongly abstract, working mainly with the semantics; another, more philosophical and applied, deals mainly with the sintax - this last is the line of research of non-classical logics (philosophically interesting) and of automated procedures, like Smullyan's semantic tableaux for proof-theory (very useful for computation theory). Today the interconnections on these areas, that were initially very close, are dangerously disappearing. Kleene's book, having been written before this separation, is much more comprehensive than the modern textbooks. About the contents: it begins with a (very well) introduction explaining the meaning of Metamathematics. Then it treats Propositional, Predicate Calculi and Formal Number Theory, written in the classical spirit that unfortunately lacks today. The third part deals with recursive functions, and the author was a first-hand researcher in the field, with many important contributions. Finally, the last part treats Model Theory as it was known then (this section can be considered pretty incomplete today).

Kleene's textbook is one of the fundamental texts of mathematical logic. It is easy to see why it is (supposedly) the most cited book in the mathematical logic literature. It is a model of clear explanation, and it does a better job of motivating the subject than any other textbook I have read (I mean deep intellectual and historical motivation of the subject, not the kind of motivation found in introductory logic books about what deduction is, and why learning logic is a good thing to do). Ishi press are to be thanked for making it readily available again at a low price. This edition however is a little on the cheaply made side; this edition was scanned from an older edition, and there are faint copy lines on most pages, so it looks like a photocopy. The pages are glued to the spine, and the binding is not flexible and does not appear all that durable. This means this edition is probably not ideal for serious study as the book will not lay out flat, and forcing it to do so may crack the spine. Nevertheless, if you are interested in mathematical logic this is a must read, and this edition makes it much easier to do so.

Mathematicians are always aware of the precision and consistency of their asserts, so they need to be trained in the very fundamentals of their science.

This book provides an enlightening vision about the basis of mathematics exploring such abstract topics as the paradoxes of set theory, transfinite numbers, and much more.

I used this book as a reference in a course I gave on mathematical logic, set theory, and the fundamentals of the number systems.

I've long owned and have read Part II in Dover copies of Stephen Kleene's 1967 Mathematical Logic text. A link to that book: Mathematical Logic That text has quite a bit in common with, and refers back to this highly influential 1952 text by him. Being able to examine and perhaps read some of this older Kleene text is a great opportunity for only about $32. Kleene was one of the main developers of recursion theory, which is there in some detail in Part III of this book. Another positive thing about this 1952 text is that it may be the first one to use the 'upside down A' modern universal quantifier symbol, which wasn't adopted by logicians in general until the 1960s. The paper book cover seems not to be notably robust, but since that cover was part of how we got access to an affordable copy of this otherwise expensive classic, I won't complain much.