Logical number theory I. An introduction
type: Item
instance of: scholarly article
title: Logical number theory I. An introduction
zbMATH Open document ID: 0759.03002
published in: Universitext
publication date: 1992-09-17T00:00:00Z
review text: This book is a most enjoyable one to read, combining the spellbinding style of a good mystery novel with meticulous coverage of technical details. The author's numerous, personable remarks on the mathematical history of various developments, excellent exercises, alternate proofs of certain important results, together with extensive reading lists and an index create a most accessible introduction to formal arithmetic, elementary recursion theory and Hilbert's tenth problem for the advanced undergraduate or beginning graduate student. By taking the time to include certain developments not usually treated, Smoryński's book may as well prove valuable to active logicians --- for example, there is a good exposition of variants of Raphael Robinson's theories \({\mathcal R}\) and \(Q\) of arithmetic, a treatment of quantifier-elimination and decidability of Skolem arithmetic (the theory of multiplication without addition), and an exposition of Dirichlet's proof of existence of integral solutions to Pell's equation \(X^ 2 -dY^ 2 = 1\) for positive non-square \(d\). Some interesting but tangential topics are treated, such as the Fueter-Pólya result showing that the only real polynomial bijective pairing function is the usual Cantor function \(\langle x,y \rangle = {{(x + y)(x + y + 1)}\over {2}} + x\) or that obtained by permutation of \(x,y\). Quite in keeping with the personable style of the book, the author at times ventures onto ground upon which few would dare to trod. For instance, pp. 193-196 describe the controversy concerning whether and to what extent credit should be given for the independent (of Matijasevič) solution of Hilbert's tenth problem with the appearance of Chudnovsky's abstract in Russian in 1970. This discussion provides the occasion for other anecdotes concerning the priority of mathematical work and may shed light on the human activity of ``doing mathematics'' for undergraduate and beginning graduate students. This particular passage on p. 196 ends wi…
Mathematics Subject Classification ID: 03-01
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 01A80
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 03-03
zbMATH DE Number: 49906
zbMATH Keywords: formal arithmetic
zbMATH Keywords: elementary recursion theory
zbMATH Keywords: Hilbert's tenth problem
author: Craig Smoryński
reviewed by: Peter Clote
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 49906