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Raymond Smullyan

Godel's incompleteness theorems

Raymond Smullyan may be known to readers of this site as the author of several books of logical puzzles. But he has also written more serious books on the subject and Godel's incompleteness theorems is such a book.

The book starts off with a look at the mathematics of self reference, leading to Tarski's theorem of the non-representability of the set of true sentences of a system. Smullyan then moves on to the proof of Gödel's first incompleteness theorem. He first does this in a system including axioms for exponentiation (following Gödel's proof) and then shows how these axioms aren't in fact necessary. The book moves on to questions of consistency of arithmetic, including ω-consistency, leading to a proof of Gödel's second incompleteness theorem. The final two chapters relate these result to the type of mathematical puzzles that Smullyan is best known for.

The book isn't for those without some previous experience in mathematical logic. The reader is rather thrown in at the deep end, and there are certainly simpler ways of presenting this material. However, it would be useful for those wanting consise but rigorous proofs of Gödel's incompleteness theorems without getting distracted by their set theoretic forms, or the details of model theory.

Product Description
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.