Introduction To Model Theory And To The Metamathematics Of Algebra

I N T R O D U C T I O N TO M O D E L THEORY A N D TO T H E METAMATHEMATICS O F ALGEBRA ABRAHAM ROBINSON University of California, Los Angeles 1963 NORTH-H 0L L A N D PUBLISH I N G COMPANY AMSTERDAM No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the publisher PRINTED I N THE NETHERLANDS PREFACE Souvienne vous de celuy B qui, mmme on demanda B quoi faire il se peinoit si fort en un art qui ne pouvoit venir B la cognoissance de gueres de gents: ”Yen ay assez de peu, respondit-il,fen ay assez d’un, j’en ay assez de pas un.” MONTAIGNE, De la Solitude The author’s book “On the Metamathematics of Algebra” which was published in this series in 1951 has now been out of print for some time. The book was concerned with the logical analysis of the methods of Abstract Algebra and beyond that, set out “to make a positive contribution to Algebra using the methods and results of Symbolic logic.” This involved among other things the development of certain topics in what is now known as Model Theory. In the years since the publication of “On the Metamathematics of Algebra” the subject has developed vigorously. Accordingly, it was decided to replace the book by an entirely new work. The result is the present volume. At an estimate, less than half of its material was given already in the earlier book, and much of this material is presented here in an different way and with a simplified terminology. With one or two exceptions, the remainder of this volume is concerned with more recent developments. The general character of the work has been changed to some extent and it should now be suitable as a textbook for a first year graduate course on the subject. Many of the topics included in “On the Metamathematics of Algebra” - such as the development of a non-countable language, the properties of classes of structures which are closed under extension or under intersection, the method of diagrams, the completeness of the notion of an algebraically closed field of given characteristic, particular applications to Algebra - are by now well - established and there is no need to justify their inclusion in the present book. On the other hand, the suggestion V VI PREFACE that numerous important concepts of Algebra possess natural generalizations within the framework of the Theory of Models has met with a rather less lively response. Nevertheless, it is still the author’s belief that investigations in this direction are both interesting and valuable. Indeed, the theory of algebraic ideals and varieties, up to the existence of the generic point for irreducible varieties, the notion of an algebraically closed extension, the notion of a system of resultants to a given set of equations, - all these can be discussed profitably in a metamathematical setting. Apart from providing a certain unity of outlook, this approach occasionally also produces new algebraic results, e.g. in the case of the concept of a differentially closed field. In the last three sections of the book, we present an introduction to Non-standard Analysis. This is a new application of Model theory which provides an effective calculus of infinitesimals and which appears to have considerable potentialities. Both the author’s working energy and the measure of patience that can be expected of a prospective reader imposed a limit on the scope of the book. Accordingly, it was not found possible to include a number of relevant items such as the theory of undecidable algebraic systems and problems (Tarski, Mostowski, R. M. Robinson, J. Robinson, Novikov, Boone, Markov, Rabin), the theory of computable algebraic systems (Frohlich-Shepherdson, Rabin, Higman), the proof-theoretic investigation of Arithmetic and Algebra (Kreisel), the properties of