Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers, Vol. 1 (Studies in Logic and the Foundations of Mathematics, Vol. 125)
Piergiorgio Odifreddi | 1992-02-18 00:00:00 | North Holland | 660 | Information Theory
1988 marked the first centenary of Recursion Theory, since Dedekind's 1888 paper on the nature of number. Now available in paperback, this book is both a comprehensive reference for the subject and a textbook starting from first principles.
Among the subjects covered are: various equivalent approaches to effective computability and their relations with computers and programming languages; a discussion of Church's thesis; a modern solution to Post's problem; global properties of Turing degrees; and a complete algebraic characterization of many-one degrees. Included are a number of applications to logic (in particular Gödel's theorems) and to computer science, for which Recursion Theory provides the theoretical foundation.
Reviews
I agree with much of what another reviewer has said (July 11, 2005 ). But I don't think that Odifreddi has exaggerated when he claims that the prerequisites of reading his book will be roughly freshman math. Having said that, I admit that since many students in their sophomore year still don't possess a working knowledge of, say proof by mathematical induction, the book is certainly too heavy for those readers. Further aggravating the difficulty for beginners is the book's admittedly less-than-ideal organization and presentation of some portion of the enormously many topics it treats. This will be frustrating for readers who are not enthusiasts of the book's multifarious viewpoints and prefer a quick introduction to the subject instead. I recommend N.Cutland's "Computability" to newcomers. Nevertheless, for those who plan a future career in mathematical logic, especially recursion theory, Odifreddi's book will prove to be a treasure trove of unusual and stimulating insights not easily encountered elsewhere - even though one might not always agree with his ideas. For the future logicians I also recommend R.Soare's "Recursively enumerable sets and degrees" for a deeper treatment of r.e. sets.
Reviews
The stated goal of the book is to have no prerequisites other than freshman math, but you really need at least undergrad logic and set theory. It is a thick book and covers a lot of interesting stuff, but there can be a frustrating lack of order and detail. He is an engaging writer, but sloppy in that he gives only part of the information you need to understand something, forgetting that he didn't give it or you don't know it. Sections are often either too sketchy or wordy and unfocused. Sometimes, though, he comes through with an elegant explanation of something. He is at his best when relating recursion theory to science, philosophy, and other branches of math. These discussions tend to be rather handwavy, but he gives references to the literature. The book as a whole is not very unified and it doesn't clearly indentify and relate the central ideas of recursion theory, so it wouldn't make a good introductory text. But it is a good reference for those with a moderate background.
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