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Introduction to Metamathematics

9 customer reviews
ISBN-13: 978-0923891572
ISBN-10: 0923891579
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Frequently Bought Together

  • Introduction to Metamathematics
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Product Details

  • Paperback: 572 pages
  • Publisher: Ishi Press (March 13, 2009)
  • Language: English
  • ISBN-10: 0923891579
  • ISBN-13: 978-0923891572
  • Product Dimensions: 6 x 1.3 x 9 inches
  • Shipping Weight: 2.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (9 customer reviews)
  • Amazon Best Sellers Rank: #440,411 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

70 of 71 people found the following review helpful By Guilherme on July 14, 2004
Format: 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).
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36 of 36 people found the following review helpful By Achronymous on November 3, 2009
Format: Paperback Verified Purchase
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.
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9 of 9 people found the following review helpful By AssemblyTimeConstant on September 2, 2012
Format: Paperback Verified Purchase
The other reviewers have lauded the book enough that any more praise may seem embarrassing to everyone involved; but more praise it deserves. It integrates some philosophy (intuitionism, formalism, and logicism) more than most modern texts, and is very thorough in its coverage. The annotated bibliography is useful, though the new introductory material added to the Isha edition seems insubstantial (e.g., the introduction cites the wikipedia article on Kleene as a source).

Some criticism it surely deserves: the lack of model theory reveals the book's age (though the reviewer Guilherme thinks this alternative perspective to be a strength). Paul E. Mokrzecki's review rather eccentrically pans the text for using the truth-functional definition of implication (we're all familiar with it: false only when the antecendent is true and the consequent false). Achronymous faults the text for its construction, but so far my copy has beautifully suffered my abuse. Sure, there are a few copy lines, but before this edition I would have had to shell out about two hundred dollars for a copy! I was heartbroken.

And another miracle has occurred: though Chang and Keisler's Model Theory may be a bit dated too (Hodges or Marker are newer, we know we know...), the Dover Publications reprint means that an affordable model theory text can accompany Kleene. The availability of cheap model theory texts makes Kleene's lack of inclusion of this subject far from disastrous.
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8 of 8 people found the following review helpful By André Gargoura on November 7, 2012
Format: Paperback
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 personal list of key books of Logic, as follows :

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.
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