Methods and Applications of Mathematical Logic Proceedings of the VII Latin American Symposium on Mathematical Logic held July 29-August 2, 1985

AMERICAn MATHEMATICAL SOCIETY VOLUME 69

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http://dx.doi.org/10.1090/conm/069

Titles in This Series Volume 38 Topics In complex analysis.

Dorothy Browne Shaffer. Editor 39 Errett Bishop: Reflections on him

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40 Integral bases for affine Lie algebras and their universal enveloping algebras. David Mitzman

41

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45

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49

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69 Methods and applications of mathematical logic. Walter A. Carnielli and Luiz Paulo de Alcantara. Editors

Methods and Applications of Mathematical Logic

COnTEMPORARY MATHEMATICS

Volume69

M'ethods and Applications of Mathematical Logic Proceedings of the VII Latin American Symposium on Mathematical Logic held July 29-August 2, 1985

Walter A. Carnielli and Luiz Paulo de Alcantara1 Editors

AMERICAn MATHEMATICAL SOCIETY Providence • RhOde Island

EDITORIAL BOARD Irwin Kra,

managing editor Thomas F. Banchoff M. Salah Baouendi Daniel M. Burns

David Eisenbud W. H. Jaco Gerald J. Janusz Jan Mycielski Alan D. Weinstein

The Proceedings of the Seventh Latin American Symposium on Mathematical Logic was held at the State University of Campinas (UNICAMP), Sao Paulo State, Brazil on July 29-August 2, 1985.

1980 Mathematics Subject Classification. (1985 Revision). Primary 01A65. 01A70. 01A99, 03A05, 03805, 03825, 03850, 03870, 03899, 03C90, 03C95, 03015, 03E10, 03E35, 03E45. 03E55. 03E65. 03F05, 03F55, 03G15, 03G20, 03G25, 03H99, 04A99. 06C15, 06F20. 06G05. 08A40. 10E05. 52A99. 60A99. 60G99. 68C25. 68060, 68065, 68T15. 68T20. 81810.

Library of Congress Cataloging-In-Publication Data

Latin-American Symposium on Mathematical Logic <7th : 1985 : State University of Campinas>

Methods and applications of mathematical logic : proceedings of the VII Latin American Symposium on Mathematical Logic held July 29-August 2, 1985 I Walter A. Carnielli and Luiz Paulo de Alcantara, editors.

p. em. -- <Contemporary mathematics, ISSN 0271-4132 ; v. 69> Sponsored by the Sociedade Brasileira de Logica, Sociedade

Brasileira de Matematica, and the Association for Symbolic Logic. Bibliography: p. ISBN 0-8218-5076-8 <alk. paper> 1. Logic, Symbolic and mathematical--Congresses. I. Carnielli,

Halter A. <Halter Alexandre> II. Alcantara, Luiz Paulo de, 1944-III. Sociedade Brasileira de Logica. IV. Sociedade Brasileira

Matematica. V. Association for Symbolic Logic. VI. Title. VII. Series: Contemporary mathematics <American Mathematical Soc i e t y l ; v. 69. QA9.A1L37 1985 511.3--dc19

Copyright @1988 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted

to the United States Government. Printed in the United States of America.

Information on Copying and Reprinting can be found at the back of this volume. This volume was printed directly from author-prepared copy.

The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. §

87-33651 CIP

CONTENTS

1 n.tJr.oduc.:Uon

Newton C.A. da Costa and L.P. de Alcantara The. -6ci.e.nti6ic. woJrk o6 A. 1. AMuda.

Section I - Foundations of set theory

Carlos A. Di Prisco and Wiktor Marek Re. 6.te.c..t..i.on pMpe.Jr..ti.u induced by -6 orne. f.altBe. c.a.Jr..cU..nai. ax.io~

A. Ehrenfeucht, M. Foreman, and J. Malitz Taxonom~c. paJr..ti..ti.o~

Akihiro Kanamori Viamond6, .taltge. c.altdi..~ and uli.Jr.a6..i.Ue.Jrt.

Section II - Algebraic logic

Marcel Guillaume Co~.tan.tu d'une. al.geb~r.e. monacUque. libJr.e. -6UJr. une. ai.gebJr.e de Boote. e..t

au.tomoJr.p~ me.-6 pllU e.Jr.van.t une. paJLti.e. geneJta.t.Jr...i.c.e de c.e.lli-ci.

Luisa Iturrioz 0Jr.de.Jr.e.d -6.tJr.uc..tuJr.u in the. du c.Jr...i.p;U.on o 6 quantum 4tJ-6.te.~:

ma.the.matic.al. pMgJr.U4

vii

ix

xi

19

27

35

45

55

viii CONTENTS

Section III - Philosophical aspects of mathematical logic

Jens E. Fenstad IY! fr.i.n,i_.ti_v., iY! mcdhe.mcdic.-6 a.rtd the. YLa.tu!r.al -6 c.ie.YLc.e..6

Andres R. Raggio The. 5oth aY!rUVe.Matty o 6 Ge.Yltze.YL' -6 thMi-6

Section IV - Interactions between logic, mathematics and computer science

Xavier Caicedo and A.r1. Sette Logie6 ar~d p-6e.udogJz.oup-6

Rolando Chuaqui Se.t-6 o 6 Jz.e.lalior~al. -6y-6te.m-6 a-6 mode.l-6 6011. -6toc.hMlic. pll.oc.e.Me.-6

Richard L. Epstein A ge.r~e.Jr.a£ 6Jz.am0Wo!z.k 6oJz. -6e.maY!tic.-6 6oJz. p1Z.opo-6ilioYLa1. logic-6

Jorg Flum FoJrmal. la.rtguagv., a.rtd topologic.al. -6pac.u

Edward H. Haeusler Automatic. the.oJz.e.m pJz.OviYLg: aY! a.tte.mpt to impJz.Ove. Jz.e.ada.biUty o 6

79

93

101

117

149

169

p1Z.oo6-6 ge.r~e.Jz.cde.d by Jz.e.-6olulioYL 179

Francisco Miraglia The. doWY!Wattd LOwe.YLhe.~-S~oR~m the.oJz.e.m 6oJz. L--6tJz.uctuJz.e.-6 iYI n--6e.t-6 189

Daniele Mundici The. de.Jrivalive. o6 tJz.uth iYL -t~Mie.wic.z -6e.Yite.Y!tial. c.al.c.ui.U-6 209

Jacques Stern Equiva1.mc.e. Jz.e.lalioY!-6 OY! la.tlic.e..6 a.rtd the. c.omple.xity o 6 :the. the.OJz.y o 6 pe.Jrmu:talio YL6 whic.h c.ormru:te.

Paulo A.S. Veloso PJz.oble.m Mlvirlg by iY!te.Jz.pJz.e.:talioY! o 6 the.oJLi.e.-6

229

241

PREFACE AND ACKNOWLEDGEMENTS

This volume constitutes the Proceedings of the Seventh Latin American Symposium on Mathematical Logic (VII LASML) which took place at the State University of Campinas (UNICAMP) in Campinas, Sao Paulo State, Brazil, from July 29 to August 2, 1985.

The meeting was co-sponsored by the Sociedade Brasileira de Logica, So-ciedade Brasileira de Matematica, and the Association for Symbolic Logic. Fi-nancial support is acknowledged from those organizations, from IBM Brasil, FAPESP {Funda~ao de Amparo a Pesquisa do Estado de Sao Paulo), CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico), Xerox do Brasil and CPFL {Companhia Paulista de For~a e Luz).

Additional financial support for the edition of this volume was provided by the Center of Logic, Epistemology and History of Science (CLE) and by re-search support funds from UNICAMP (FAP- Fundo de Apoio a Pesquisa).

The VII Latin American Symposium on Mathematical Logic follows in the tradition begun in 1970 when due to the efforts of Rolando Chuaqui and the Advisory Committee for Logic in Latin America, under the auspices of the As-sociation for Symbolic Logic, the I LASML was held in Santiago, Chile.

Subsequently there have been meetings in Brasilia (II LASML, 1972), Cam-pinas (III LASML, 1976), Santiago (IV LASML, 1978), Bogota (V LASML, 1980) and Caracas (VI LASML, 1983).

The history of the LASML's reflects the strugle in Latin America against the financial problems, political instability, geographical and other diffi-culties which explains in part the odd sequence of dates of the Symposia.

However, the field of mathematical logic is well and alive in Latin Ame-rica, as showed by the vitality of the publications which have resulted from the meetings:

ix

X PREFACE AND ACKNOWLEDGEMENTS

- Non-Cfus~-<-c.a£ Lag-<-c.~, Madel Theo!ty and Computab-<-lLty, edited by A. I. Arruda, N.C.A. da Costa and R. Chuaqui, series Studies in Logic and the Foun-dations of Mathematics, North Holland Pub. Co., 1977 (proceedings of the III LASt1L.).

- Matltema:t-<-c.al Lag-<-c. -<-n La.D.n Ame!Uc.a (edited by A. I. Arruda, R. Chuaqui and N.C.A. da Costa, series Studies in Logic and the Foundations of Mathemat-

ics, North Holland Pub. Co., 1979 (proceedings of the IV LASML).

-Method~ i.n Ma:thematic.al Log.Zc., edited by C.A. Di Prisco, series Lectu-re Notes in Mathematics 1130, Springer-Verlag, 1983 (proceedings of the VI LASML).

In the periods between the Latin American Symposia the Brazilian group organizes the Brazilian Conference on Mathematical Logic (BCML) though suffering from similar difficulties, has had enough support to in publications such as:

logic which, result

- Ma:thema:tic.al Lag-<-c., edited by A. I. Arruda, N.C.A. da Costa and R.Chua-qui, series Lecture Notes in Pure and Applied Mathematics volume 39, Marcel Dekker Inc., 1978 (proceedings of the I BCML).

- Pitoc.c.ed-<-ng6 a 6 the Th-<-~t.d BJwz-<-.l'iavt Cc rJ 6eJtc.nc.e on Mcuhe.ma:t-<-c.a£ Logic.,

edited by A.I. Arruda, N.C.A. da Costa and A.M. Sette, Sociedade Brasileira de Logi ca 1980 (proceedings of the III BCML).

The organizing committee for the present conference included logicians from Argentina, Brazil, Chile, Colombia and Venezuela.

We are indebted to Mineko Yamashita and Celina A.A.P. Abar for their collaboration in the composition and proofreading of the material, and toRi-chard Epstein who contributed with his experience in writting and organ1z1ng books. l~e are also grateful to the Af.:S staff, in the person of t1s. t~ary C. Lane, and to Professor Irwin Kra, of the Editorial Board of Mathematics for this opportunity of joint work.

Contemporary

The symposium was dedicated to the memory of Professor Ayda Ignes Arruda (June 27, 1936- October 13, 1983), in acknowledgment of her efforts towards developing mathematical logic in Latin America and especially in Brazil. She was able to accomplish that because of the energetic work she devoted to or-ganizing and making better known the field of logic, and because she was es-teemed for her scientific work, which is reported on in the first paper.

INTRODUCTION

The title of this volume illustrates one of the interesting points about the work developped in the field of logic in Latin America, namely its con-nections to other fields, especially to philosophy, computer science and prob-ability theory, and the interest in Latin America in nonclassical logics. This is particularly important since areas such as artificial intelligence, software engineering, mathematical biology, physics, and linguistics appear now to need mathematical methods of the kind studied in logic.

The papers appearing in this volume represent revised versions Of most of the lectures delivered at the conference Abstracts of all the addresses presented, including short communications and contributions by title, appear-ed in The Journal of Symbolic Logic vol. 31, number 4, pp. 1093-1103, 1986.

The volume is divided into four sections: Foundations of Set Theory, Al-gebraic Logic, Philosophical Aspects of Mathematical Logic and Interactions between Logic, Mathematics and Computer Science. Not by coincidence, the fourth section contains the majority of the papers, leading evidence· to this tendency towards interdisciplinarity which we, as organizers, merely report.

xi

Walie.Jt Ale.xand.lle. Ca.lll1.(.e.lli Luiz Pauto de AlcantaJta

Institute of Mathematics, Statistics and Computer Science State University of Campinas

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