Introduction to Metamathematics [Paperback]

Stephen Cole Kleene (Author), Michael Beeson (Foreword)

Book Description

ISBN-10: 0923891579 | ISBN-13: 978-0923891572 | Publication Date: March 13, 2009

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Gödel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic, at least a turning point after which “nothing was ever the same.” Kleene was an important figure in logic, and lived a long full life of scholarship and teaching. The 1930s was a time of creativity and ferment in the subject, when the notion of “computable” moved from the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Göde1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of “computable”. When they all turned out to be equivalent, there was a collective realization that this was indeed the “right notion”. Kleene played a key role in this process. One could say that he was “there at the beginning” of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Gödel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.

Product Details

• Paperback: 572 pages
• Publisher: Ishi Press (March 13, 2009)
• Language: English
• ISBN-10: 0923891579
• ISBN-13: 978-0923891572
• Product Dimensions: 9 x 6 x 1.6 inches
• Shipping Weight: 1.9 pounds (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #446,746 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

48 of 49 people found the following review helpful:

5.0 out of 5 stars The classic of the classics, July 14, 2004

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Guilherme (Săo Paulo, Brazil) - See all my reviews

This review is from: Introduction to Metamathematics (Bibliotheca Mathematica) (Hardcover)

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).

16 of 16 people found the following review helpful:

5.0 out of 5 stars Five stars for content, three and a half stars for the new edition, November 3, 2009

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Achronymous - See all my reviews

Amazon Verified Purchase

This review is from: Introduction to Metamathematics (Paperback)

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.

22 of 31 people found the following review helpful:

5.0 out of 5 stars Enlightening reading, March 23, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: Intro to Metamathematic (Bibliotheca Mathematica, a Series of Monographs on Pure and) (Hardcover)

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.