ANDRZEJ MOSTOWSKI AND FOUNDATIONAL STUDIES This page intentionally left blank Andrzej Mostowski and Foundational Studies Edited by A. Ehrenfeucht Department of Computer Science, University of Colorado, CO, USA V.W. Marek Department of Computer Science, University of Kentucky, KY, USA and M. Srebrny Institute of Computer Science, Polish Academy of Sciences, Poland Amsterdam • Berlin • Oxford • Tokyo • Washington, DC © 2008 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-782-6 Library of Congress Control Number: 2007939570 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Distributor in the USA and Canada Gazelle Books Services Ltd. IOS Press, Inc. White Cross Mills 4502 Rachael Manor Drive Hightown Fairfax, VA 22032 Lancaster LA1 4XS USA United Kingdom fax: +1 703 323 3668 fax: +44 1524 63232 e-mail: [email protected] e-mail: [email protected] LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS Andrzej Mostowski and Foundational Studies v A. Ehrenfeucht, V.W. Marek and M. Srebrny (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. Preface Andrzej Mostowski was one of the world’s leading European scientists, preserving and expanding a national school of research: in his case the Polish School of Logic. This effort took place within the adverse political and social circumstances of post-World War II Warsaw. Having taught at the underground Warsaw University during the War, Mostowski was active in rebuilding the teaching and research infrastructure in Poland, based on the tradition of logic and mathematics research that had already led to the coun- try’s outstanding reputation before the War. The country’s world-wide contribution to research and teaching in logic, mathematics and computer science after the War bear witness to the impact that this Logic School, Professor Mostowski, his collaborators and students have had. Apart from being a key person in teaching and research management, Mostowski was a profound researcher with contributions in the main areas of mathemat- ical logic and the foundations of mathematics. This volume is a collection of fifteen research and expository articles, a complete bibliography of Andrzej Mostowski’s writings, three biographical and historical articles, and eleven short personal reminiscences, all aimed at illuminating Andrzej Mostowski’s ideas and personality. Our main motivation for soliciting and editing these articles was that the name “Mostowski” arouses a great deal of favorable interest in very many people from various countries and in many different research communities. A starting point for anyone interested in the work of Mostowski is his famous se- ries of lectures “Thirty years of foundational studies”, published in 1965, considered by many as an ultimate presentation of the then-current state of mathematical logic, and more generally, the foundations of mathematics. Michael Dunn (logician and computer scientist, Indiana University, Bloomington) told us that he came across “Thirty years ...” as a student in Pittsburgh and found the work so interesting and beautiful that he carried it everywhere for the next two years, trying to convince everyone around him to read it. It is not clear whether Mostowski would be happy with the title of the current col- lection. Modest as he was, he could very well argue that the scope of the foundations of mathematics is too large these days to be encompassed by a single volume and a single researcher. The explosion of foundational research resulting from challenges and progress in computer science and artificial intelligence makes it almost impossible to present even their main ideas in a single book. Their impact is even felt further afield, in economics, social science, engineering, cognitive sciences and philosophy. The contributions included in this collection present a variety of approaches to dif- ferent areas of logic and the foundations of mathematics. They are written on different levels, by various researchers presenting a variety of views. We believe that these articles can continue to stimulate research in the areas pioneered and mastered by Mostowski. Reflecting Mostowski’s diverse interests, the research contributions in this volume relate to many distinct areas. Balcar and Jech study a class of Boolean algebras and its relationship with tech- niques used in set theory, in particular forcing. vi Dickmann and Petrovich exhibit an important interaction between three-valued logic and the algebraic theory of diagonal quadratic forms over a kind of rings, called semi- real. Relating to the original studies of Mostowski in the late 40’s and 50’s of the 20th century, Friedman discusses the current progress in Gödel phenomena, mainly incom- pleteness. Friedman lays out a program of significant progress that needs to be made in our comprehension of the scope of incompleteness. The paper challenges the foun- dational community with specific questions that need to be studied to achieve a deeper understanding of incompleteness. Grzegorczyk and Zdanowski develop basic decidability results without resorting to Gödel’s encoding of formulas as integers. Instead, they work with the elementary theory of strings, introduced by Tarski under the name of “concatenation” in his celebrated paper on the concept of truth. They prove that the theory of strings is essentially undecidable. Guzicki and Krynicki provide a new result in the area of generalized quantifiers. They define and pursue a quantifier that does not meet the very general definition of a quantifier proposed by Mostowski in 1957. Keisler extends certain results of Mostowski on the complexity class of the concept of limz→∞ F (z)=∞, showing that in many structures the limit cannot be defined with fewer than three quantifier blocks, but in some more powerful structures the limit property for arbitrary functions can be defined in both two-quantifier forms. Knight presents the current state of research in the areas of the arithmetic (Kleene– Mostowski) and hyperarithmetic (Davis–Mostowski) hierarchies. In recent years, re- search in this area has merged with effective aspects of model theory, resulting in a va- riety of deep results on the complexity of algebraic structures and constructions. The current research in this area, sometimes called “recursive mathematics” is presented. Kotlarski deals with the interplay of syntactic and semantic arguments in Peano Arithmetic and its fragments. His paper outlines significant progress in the study of for- malized arithmetic since Mostowski’s death. The results of these investigations have re- vised our understanding of the nature of independent formulas in Peano Arithmetic, ex- hibiting sentences that shattered the conviction (current in some mathematical circles) that “ordinary combinatorial sentences” cannot be independent from Peano Arithmetic. A Ramsey-like sentence of this sort was discovered by Paris and Harrington. Numerous other examples were found by Friedman and others. Makowsky deals both with the reminiscences of Mostowski in the social con- text of doing mathematics, and with specific mathematical developments grounded in Mostowski’s work. Those developments, particularly in model theory and in abstract model theory, where Makowsky contributed significantly, have found application, out- side pure mathematics, in theoretical computer science. Murawski and Wolenski´ investigate the philosophy of mathematics espoused in Mostowski’s papers. Mostowski’s work was firmly grounded in the philosophical tradi- tion of his teachers, including Tarski and Kotarbinski.´ The authors show how Mostowski related to the foundational problems in many of his contributions, especially in his work and presentations of Gödel’s incompleteness and Cohen’s independence results. Mycielski investigates the issue of mathematical truth, its formal and informal as- pects, and its relationship with mathematical practice. The paper discusses Tarski’s the- ory of truth and the question of the presence of the hierarchy of metatheories in everyday mathematics, metatheories determined by some bounds on the allowed lengths of proofs. vii L’Innocente and Macintyre present a very nice application of a model-theoretic result of Mostowski on direct products of models to certain fundamental problems of a special Lie algebra and the Diophantine geometry of curves. The two expository papers by Scott and by Jankowski and Skowron are devoted to the algebraic approach to formal logic. Scott focuses on the quantifiers interpreted as the infs and sups, respectively, in the algebras interpreting various logics. Jankowski and Skowron sketch the past and present perspective of the role of algebraic logic and Pawlak’s rough sets in artificial intelligence. Wells discusses the issue of pseudorecursive varieties, that is, varieties V of algebras such that for each natural number n, the set of equations with at most n variables true in all algebras of V is recursive, but such that the set of equations true in all algebras of V is not recursive. The paper outlines progress in the area as well as a number of important problems. Some of the contributors pointed out to us that their topics and results are derived from Alfred Tarski’s inspiration. We quickly agreed to underline the more general per- spective of the Tarski/Mostowski Berkeley/Warsaw school. The reader will find refer- ences to Tarski’s work, ideas and inspirations in several contributions in this volume. The work and vision of Alfred Tarski, albeit from afar after the WWII, significantly in- fluenced both Mostowski’s research and that of his collaborators. It is natural to treat Mostowski as “the Prince of the Tarskian kingdom”, as B.F. Wells, the last of Tarski’s students, does in his contribution. (In spite of many efforts we have not been able to trace any letter of the postal correspondence between Tarski and Mostowski.