CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 Logic Colloquium 2006 The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2006, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. The most topical areas of current research are covered: valued fields, Hrushovski constructions (from model theory), algorithmic randomness, relative computability (from computabil- ity theory), strong forcing axioms and cardinal arithmetic, and large cardinals and determinacy (from set theory), as well as foundational topics such as algebraic set theory, reverse mathematics, and unprovability. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic. i CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 ii CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 LECTURE NOTES IN LOGIC A Publication of The Association for Symbolic Logic This series serves researchers, teachers, and students in the field of symbolic logic, broadly interpreted. The aim of the series is to bring publications to the logic community with the least possible delay and to provide rapid dissemina- tion of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board Anand Pillay, Managing Editor Department of Pure Mathematics, School of Mathematics, University of Leeds Lance Fortnow Department of Computer Science, University of Chicago Shaughan Lavine Department of Philosophy, The University of Arizona Jeremy Avigad Department of Philosophy, Carnegie Mellon University Vladimir Kanovei Institute for Information Transmission Problems, Moscow Steffen Lempp Department of Mathematics, University of Wisconsin See end of book for a list of the books in the series. More information can be found at http://www.aslonline.org/books-lnl.html iii CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 iv CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 Logic Colloquium 2006 Edited by S. BARRY COOPER University of Leeds HERMAN GEUVERS Radboud University Nijmegen ANAND PILLAY University of Leeds JOUKO VA¨ AN¨ ANEN¨ University of Amsterdam association for symbolic logic v CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521110815 © Association for Symbolic Logic 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-60502-4 eBook (NetLibrary) ISBN-13 978-0-521-11081-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 CONTENTS Introduction .................................................... ix Marat M. Arslanov Definability and elementary equivalence in the Ershov difference hierarchy .................................................... 1 Benno van den Berg and Ieke Moerdijk A unified approach to algebraic set theory . ..................... 18 Andrey Bovykin Brief introduction to unprovability ............................. 38 Venanzio Capretta and Amy P. Felty Higher-order abstract syntax in type theory ...................... 65 Raf Cluckers An introduction to b-minimality................................ 91 Rod Downey The sixth lecture on algorithmic randomness .................... 103 Harvey M. Friedman The inevitability of logical strength: Strict reverse mathematics . 135 Martin Goldstern Applications of logic in algebra: Examples from clone theory . 184 Ehud Hrushovski On finite imaginaries.......................................... 195 Andrew E. M. Lewis Strong minimal covers and a question of Yates: The story so far . 213 Antonio Montalb´an Embeddings into the Turing degrees ............................ 229 Jan Reimann Randomness—beyond Lebesgue measure....................... 247 vii CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 viii CONTENTS J. R. Steel The derived model theorem .................................... 280 Boban Veliˇckovi´c Forcing axioms and cardinal arithmetic ......................... 328 Frank O. Wagner Hrushovski’s amalgamation construction........................ 361 CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 INTRODUCTION These are the proceedings of the Logic Colloquium 2006, which was held July 27–August 2 at the Radboud University of Nijmegen in the Netherlands. The Logic Colloquium is the annual European conference on logic, orga- nized under the auspices of the Association for Symbolic Logic (ASL). The program of LC2006 consisted of a mixture of tutorials, invited plenary talks, special sessions, and contributed talks. Finally, there was a plenary discussion on G¨odel’s legacy, on the occasion of the 100th birthday of the great logician Kurt G¨odel, moderated by William Tait. The program gave a good overview of the recent research developments in logic. The tutorial speakers were Downey, Moerdijk, and Veliˇckovi´c.The invited plenary speakers were Abramsky, Arslanov, Friedman, Goldstern, Hrushovski, Koenigsmann, Lewis, Montalb´an,Palmgren, Pohlers, Schimmerling, Steel, Tait, and Wagner. The five special sessions were devoted to computability the- ory, computer science logic, model theory, proof theory and type theory, and set theory. For these proceedings we have invited the tutorial and plenary invited speakers—as well as one invited speaker from each of the special sessions—to submit a paper. All papers have been reviewed by independent referees. This has given rise to these proceedings, which give a good overview of the content and breadth of the Logic Colloquium 2006 and of the state of the art in logic at present. The Editors S. Barry Cooper Herman Geuvers Anand Pillay Jouko V¨a¨an¨anen ix CUUS603-cooper July 11, 2009 8:55 978 0 521 11081 5 x DEFINABILITY AND ELEMENTARY EQUIVALENCE IN THE ERSHOV DIFFERENCE HIERARCHY M. M. ARSLANOV Abstract. In this paper we investigate questions of definability and elementary equivalence in the Ershov difference hierarchy. We give a survey of recent results in this area and discuss a number of related open questions. Finally, properties of reducibilities which are intermediate between Turing and truth table reducibilities and which are connected with infinite levels of the Ershov hierarchy are studied. §1. Introduction. In this paper we consider the current status of a number of open questions concerning the structural organization of classes of Turing degrees below 0 , the degree of the Halting Problem. We denote the set of all such degrees by D(≤ 0). The Ershov hierarchy arranges these degrees into different levels which are determined by a quantitative characteristic of the complexity of algorithmic recognition of the sets composing these degrees. The finite level n, n ≥ 1, of the Ershov hierarchy constitutes n-c.e. sets which can be presented in a canonical form as − [ n 1 ] 2 A = (R2i+1 − R2i ) ∪ (R2i − R2i+1) i=0 for some c.e. sets R0 ⊆ R1 ⊆ R2 ⊆ ··· ⊆ Rn−1. (Here if n is an odd number then Rn = ∅.) A (Turing) degree a is called an n-c.e. degree if it contains an n- c.e. set, and it is called a properly n-c.e. degree if it contains an n-c.e. set but no (n − 1)-c.e. sets. We denote by Dn the set of all n- c.e. degrees. R denotes the set of c.e. degrees. Degrees containing sets from different levels of the Ershov hierarchy, in particular the c.e. degrees, are the most important representatives of D(≤ 0). Investigations of these degree structures pursued in last two-three decades The author is supported by RFBR Grant 05-01-00830. Logic Colloquium ’06 Edited by Barry Cooper, Herman Geuvers, Anand Pillay, and Jouko Va¨an¨ anen¨ Lecture Notes in Logic, 32 c 2009, Association for Symbolic Logic 1 2 M. M. ARSLANOV show that the c.e. degrees and the degrees from finite levels of the Ershov hierarchy have similar properties in many respects. The following theorem, which states that the classes of c.e. and n-c.e. degrees for 1 <n<are indistinguishable from the point of view of their ability to compute fixed-point free functions, is a remarkable confirmation of this observation. Theorem 1. (Arslanov [1] for n =1; Jockusch, Lerman, Soare, Solovay [17] for n>1). Suppose that A is a set which is n-c.e. for some n ≥ 1.ThenA has degree 0 if and only if there is a function f computable in A with no fixed point, ∀ i.e. ( e)(Wf(e) = We). Nevertheless, the elementary theories of the c.e. and the n-c.e. degrees for 0 every n>1 are different even at the Σ2-level. This was shown by Downey [13] (the diamond lattice is embeddable in 2-c.e. degrees preserving 0 and 0)and later by Cooper, Harrington, Lachlan, Lempp and Soare [10] (there is a 2-c.e. degree d < 0 which is maximal in the partial ordering of all n-c.e. degrees for all n>1). Previous to this, a difference between the elementary theories of 0 these degree structures at the Σ3-level was exhibited in Arslanov [2, 3] (every n-c.e. degree for any n>1 can be cupped to 0 bya2-c.e.degree< 0). (Since 0 R D any Σ1-sentence satisfies and n for any n>1 if and only if it is consistent with the theory of partial orderings, R and Dn are indistinguishable at the 0 Σ1-level.) These results initiated an intensive study of the properties of the n-c.e.