Introduction to Metamathematics (Bibliotheca Mathematica) (英語) ハードカバー – 1980/1/1
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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 Gadel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic, at least a turning point after which œnothing 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 â€œcomputableâ€ moved from the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gade1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of â€œcomputableâ€. When they all turned out to be equivalent, there was a collective realization that this was indeed the œright notion. Kleene played a key role in this process. One could say that he was œthere at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Gadel'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.
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.
There are other books which also present the same negative and positive answers to the arithmetic completeness question, but this book by Kleene presents a very thorough basis in propositional and predicate calculus along the way, for which metamathematical theorems are developed towards obtaining these answers.
On page 82, a more or less Hilbert-style formal system is given in three parts: first a set of propositional calculus axioms and one rule (modus ponens), then a set of predicate calculus axioms and two rules (universal introduction and existential elimination), and finally a set of non-logical axioms for a first-order language with equality including arithmetic addition and multiplication. This system is developed in Kleene's particular style until Gödel's theorem can be stated and proved (minus one lemma) on pages 204-213.
Pages 217-439 present a broad range of mathematical logic, including recursive functions, the Gödel numbering, Post's theorem, Church's theorem, computable functions, and Skolem's paradox.
Finally Chapter XV (pages 440-516) present Gentzen's system in much the same form as Gentzen's original 1934 paper, within which the consistency of number theory can be proved. (See pages 476-479.)