Selected logic papers

SELECTED LOGIC PAPERS

World Scientific Series in 20th Century Mathematics

Published Vol. 1

The Neumann Compendium edited by F. Brddy and T. Vamos

Vol. 2

40 Years in Mathematical Physics by L D. Faddeev

Vol. 3

Selected Papers of Yu I Manin by Y. Manin

Vol. 4

A Mathematician and His Mathematical Work — Selected Papers of S. S. Chem edited by S. Y. Cheng, P. Li and G. Tian

Vol. 5

Fields Medallists' Lectures edited by Sir Michael Atiyah and Daniel lagolnitzer

Vol. 6

Selected Logic Papers by G. E. Sacks

World Scientific Series in 20th Century Mathematics - Vol. 6

SELECTED LOGIC PAPERS

Gerald E. Sacks Department of Mathematics Harvard University and Massachusetts Institute of Technology USA

World Scientific Singapore 'New Jersey •London ■ Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Sacks, Gerald E. Selected logic papers / Gerald E. Sacks. p. cm. -- (World Scientific series in 20th century mathematics; vol. 6) Includes bibliographical references. ISBN 9810232675 (alk. paper) 1. Logic, symbolic and mathematical. I. Title. II. Series. QA9.2.S23 1999 511.3-dc21 99-22812 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

The author and publisher would like to thank the following publishers of the various journals and books for their assistance and permission to include the selected reprints found in this volume: Academic Press American Mathematical Society Association for Symbolic Logic Elsevier Science Publishers Gauthier-Villars Johns Hopkins University Press Polish Academy of Sciences Princeton University Press While every effort has been made to contact the publishers of reprinted papers prior to publication, we have not been successful in some cases. Where we could not contact the publishers, we have acknowledged the source of the material. Proper credit will be accorded to these publishers in future editions of this work after permission is granted.

Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic oo mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by Uto-Print.

In Memory Of My Mother Ethel Sacks

This page is intentionally left blank

vil

Contents

General Introduction

ix

[I]

On the Degrees Less Than Q\Ann. Math. 77 (1963) 211-231

1

[2]

Recursive Enumerability and the Jump Operator, Trans. Amer. Math. Soc. 108 (1963) 223-239

22

The Recursively Enumerable Degrees are Dense, Ann. Math. 80 (1964)300-312

39

A Simple Set Which is Not Effectively Simple, Proc. Amer. Math. Soc. 15(1964)51-55

52

Metarecursive Sets (with Georg Kreisel), J. Symbolic Logic 30(1965)318-338

57

Post's Problem, Admissible Ordinals, and Regularity, Trans. Amer. Math. Soc. 124 (1966) 1-23

78

[3]

[4]

[5]

[6]

[7]

[8]

[9]

On a Theorem of Lachlan and Martin, Proc. Amer. Math. Soc. 18(1967)140-141

101

A Minimal Hyperdegree (with Robin O. Gandy)) Fundamenta Math. 61(1967)215-223

103

Measure-Theoretic Uniformity in Recursion Theory and Set Theory, Trans. Amer. Math. Soc. .42 21969) )31-420

[10] Recursion in Objects of Finite Type, Proc. Int. Congress Math. (1970), Vol. 1,251-254 [II]

Forcing with Perfect Closed Sets, Proc. Symp. Pure Math.. Vol. 13, Part 1, Axiomatic Set Theory (1971) 331-355

[ 12] The a-Finite Injury Method (with Stephen G. Simpson), Ann. Math. Logic 4 (1972)343-367

111

152

156

181

Vlll

[ 13] The 1 -Sectton of a Type n Objectt Generalized Recursion Theory (edited by J. E. Fenstad and P. G. Hinman) (1974) 81-93

206

[14] Remarks Against Foundational Activity, Historia Math. 2 (1975) 523-528

219

[15] Countable Admissible Ordinals and Hyperdegrees, Adv. Math. 20(1976)213-262

225

[ 16] Thefc-Secttonof a Type n Object, Amer. J. Math. 99 (1977) 901-917

275

[17] Effective Bounds on Morley Rank, Fundamenta Math. 53 (1979) 111-121

292

[18] Post's Problem, Absoluteness and Recursion in Finite Types, The Kleene Symposium (edited by J. Barwise, H. J. Keisler and K. Kunen) (1980) 201 -222

303

[19] On the Number of Countable Models, Southeast Asian Conference on Logic (edited by C.-T Chong and M. J. Wicks) (1983)185-195

325

[20] Post's Problem in ^-Recursion, Proc. Symp. Pure Math., Vol. 42 (1985)177-193

336

[21 ] The Limits of ^-Recursive Enumerability, Ann. Pure Appl. Logic 31(1986)87-120

353

[22] Inadmissible Forcing (with T A. Slaman), Adv. Math. 66 (1987) 1-30

[23] Effective Forcing Versus Proper Forcing, Ann. Pure Appl. Logic 81(1996)171-185

387

417

ix

General Introduction

Nothing

is forgotten,

fortunately

or not. -H. M.

Looking backward I see a problem solver more than a system builder and still do. I always felt more comfortable working on someone else's problem. Avoidance of some sort I suppose. Two exceptions come to mind immediately, splitting an re set into incomparable re s e t s [ l ] and solving Post's problem invariantly. The first was my initial foray into re sets and I am still working on the second. T h e problem with problem solving is being unable to let go of an intractable problem. This did not happen to me, fortunately, until fairly late in the game, about fifteen years ago I think. The papers I selected for reprinting in this volume are more interesting to me t h a n the ones I left out. I do not know if there is a common theme. For me logic is about definability, but t h a t could be a recursion theorist talking. On the other hand the principal methods of recursion theory, priority arguments and forcing, do apply in other areas of mathematical logic. a a

T . Slaman put it as follows. "There is a particular perspective obtained by consistently including an understanding of 'definability' in the solution to a mathematical problem. It is the assumption of that vantage point which I identify with recursion theory. Of course not only 'recursion theorists', but also computer scientists, descriptive set theorists and others have claim to it; many of us ascend but we do not all look in the same direction. We are all trying to understand the interaction between mathematical objects and the means needed to speak .about, them.!'., , . , F

Copyngnted Material

X

On the degrees less than 0' [1] appeared in 1963. T h e paper presents two results from my Ph.D. thesis (Cornell 1961) directed by J. B. Rosser. T h e first is a splitting theorem for re sets with several corollaries. My favorite then and now is: each degree strictly between 0 and 0' is incomparable with some re degree. The second is the existence of a minimal degree below 0'. I was intrigued by the idea of using the priority method to construct a non-re degree, an approach taught to me by Shoenfield [rl]. T h e paper is devoid of intuitive remarks save for the final paragraph. It is tempting to blame the lack on Kleene's influence, but the t r u t h is that in 1961 I greatly distrusted intuitive remarks and regarded them as intellectually corrupting. These days formal reasoning seems too weak to stand alone, like a skeleton without a body. Recursive enumerability and the jump operator [2] was written in the fall of 1961. The main result, a determination of the range of the j u m p restricted to re degrees, was inspired by Friedberg's determination of the range of the j u m p [r2], and by Shoenfield [rl] on the j u m p restricted to degrees below 0'. This paper in harmony with its predecessor was free of intuitive remarks. The search for the proof led me to a simple instance of the infinite injury method. Phrases such as "finite injury" or "infinite injury" did not occur. The recursiveyy enumerable degrees are dense [3] needed something more than infinite injury, an insight that was painfully long in coming. Density says t h a t if B and C are re sets such t h a t B