Admissible Sets and Structures 9781316717196

Table of contents :
Table of Contents......Page 12
Introduction......Page 16
Part A. The Basic Theory......Page 20
1. The Role of Urelements......Page 22
2. The Axioms of KPU......Page 24
3. Elementary Parts of Set Theory in KPU......Page 26
4. Some Derivable Forms of Separation and Replacement......Page 29
5. Adding Defined Symbols to KPU......Page 33
6. Definition by Σ Recursion......Page 39
7. The Collapsing Lemma......Page 45
8. Persistent and Absolute Predicates......Page 48
9. Additional Axioms......Page 53
1. The Definition of Admissible Set and Admissible Ordinal......Page 57
2. Hereditarily Finite Sets......Page 61
3. Sets of Hereditary Cardinality Less Than a Cardinal k......Page 67
4. Inner Models: the Method of Interpretations......Page 69
5. Constructible Sets with Urelements; HYPm Defined......Page 72
6. Operations for Generating the Constructible Sets......Page 77
7. First Order Definability and Substitutable Functions......Page 84
8. The Truncation Lemma......Page 87
9. The Levy Absoluteness Principle......Page 91
1. Formalizing Syntax and Semantics in KPU......Page 93
2. Consistency Properties......Page 99
3. M-Logic and the Omitting Types Theorem......Page 102
4. A Weak Completeness Theorem for Countable Fragments......Page 107
5. Completeness and Compactness for Countable Admissible Fragments......Page 110
6. The Interpolation Theorem......Page 118
7. Definable Well-Orderings......Page 120
8. Another Look at Consistency Properties......Page 124
1. On Set Existence......Page 128
2. Defining Π11 and Σ11 Predicates......Page 131
3. Π11 and Δ11 on Countable Structures......Page 137
4. Perfect Set Results......Page 142
5. Recursively Saturated Structures......Page 152
6. Countable M-Admissible Ordinals......Page 159
7. Representability in M-Logic......Page 161
PartB. The Absolute Theory......Page 166
1. Satisfaction and Parametrization......Page 168
2. The Second Recursion Theorem for KPU......Page 171
3. Recursion Along Well-founded Relations......Page 173
4. Recursively Listed Admissible Sets......Page 179
5. Notation Systems and Projections of Recursion Theory......Page 183
6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals......Page 188
7. Ordinal Recursion Theory: Stability......Page 192
8. Shoenfield's Absoluteness Lemma and the First Stable Ordinal......Page 204
1. Inductive Definitions as Monotonic Operators......Page 212
2. Σ Inductive Definitions on Admissible Sets......Page 220
3. First Order Positive Inductive Definitions and IHYPm......Page 226
4. Coding IHFm on M......Page 235
5. Inductive Relations on Structures with Pairing......Page 245
6. Recursive Open Games......Page 257
Part C. Towards a General Theory......Page 270
1. Some Definitions and Examples......Page 272
2. A Weak Completeness Theorem for Arbitrary Fragments......Page 277
3. Pinning Down Ordinals: the General Case......Page 285
4. Indiscernibles and upward Lowenheim-Skolem Theorems......Page 291
5. Partially Isomorphic Structures......Page 307
6. Scott Sentences and their Approximations......Page 312
7. Scott Sentences and Admissible Sets......Page 318
1. The König Infinity Lemma......Page 326
2. Strict Π11 Predicates: Preliminaries......Page 330
3. König Principles on Countable Admissible Sets......Page 336
4. König Principles K1 and K2 on Arbitrary Admissible Sets......Page 341
5. König's Lemma and Nerode's Theorem: a Digression......Page 349
6. Implicit Ordinals on Arbitrary Admissible Sets......Page 354
7. Trees and Σ1 Compact Sets of Cofinality ω......Page 358
8. Σ1 Compact Sets of Cofinality Greater than ω......Page 367
9. Weakly Compact Cardinals......Page 371
1. Compactness Arguments over Standard Models of Set Theory......Page 380
2. The Admissible Cover and its Properties......Page 381
3. An Interpretation of KPU in KP......Page 387
4. Compactness Arguments over Nonstandard Models of Set Theory......Page 393
References......Page 394
Index of Notation......Page 401
Subject Index......Page 403

Citation preview

Admissible Sets and Structures Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Admissible set theory is a major source of interaction between model theory, recursion theory and set theory, and plays an important role in definability theory. In this volume, the 7th publication in the Perspectives in Logic series, Jon Barwise presents the basic facts about admissible sets and admissible ordinals in a way that makes them accessible to logic students and specialists alike. It fills the artificial gap between model theory and recursion theory and covers everything the logician should know about admissible sets. JON BARWISE

Madison.

works in the Department of Mathematics at the University of Wisconsin,

PERSPECTIVES IN LOGIC

The Perspectives in Logic series publishes substantial, high-quality books whose central theme lies in any area or aspect of logic. Books that present new material not now available in book form are particularly welcome. The series ranges from introductory texts suitable for beginning graduate courses to specialized monographs at the frontiers of research. Each book offers an illuminating perspective for its intended audience. The series has its origins in the old Perspectives in Mathematical Logic series edited by the ^2-Group for "Mathematische Logik" of the Heidelberger Akademie der Wissenschaften, whose beginnings date back to the 1960s. The Association for Symbolic Logic has assumed editorial responsibility for the series and changed its name to reflect its interest in books that span the full range of disciplines in which logic plays an important role. Arnold Beckmann, Managing Editor Department of Computer Science, Swansea University Editorial Board: Michael Benedikt Department of Computing Science, University of Oxford Elisabeth Bouscaren CNRS, Département de Mathématiques, Université Paris-Sud Steven A. Cook Computer Science Department, University of Toronto Michael Glanzberg Department of Philosophy, University of California Davis Antonio Montalban Department of Mathematics, University of Chicago Simon Thomas Department of Mathematics, Rutgers University For more information, see www.aslonline.org/books_perspectives.html

PERSPECTIVES IN LOGIC

Admissible Sets and Structures An Approach to Definability Theory

JON BARWISE University of Wisconsin, Madison

ASSOCIATION f o r s y m b o l i c l o g i c

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University's mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107168336 10.1017/9781316717196 First edition © 1975 Springer-Verlag Berlin Heidelberg This edition © 2016 Association for Symbolic Logic under license to Cambridge University Press. Association for Symbolic Logic Richard A. Shore, Publisher Department of Mathematics, Cornell University, Ithaca, NY 14853 http ://ww w. aslonline. org This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. A catalogue record for this publication is available from the British Library. ISBN 978-1-107-16833-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

To my mother and the memory of my father

Preface to the Series

On Perspectives. Mathematical logic arose from a concern with the nature and the limits of rational or mathematical thought, and from a desire to systematise the modes of its expression. The pioneering investigations were diverse and largely autonomous. As time passed, and more particularly in the last two decades, interconnections between different lines of research and links with other branches of mathematics proliferated. The subject is now both rich and varied. It is the aim of the series to provide, as it were, maps or guides to this complex terrain. We shall not aim at encyclopaedic coverage; nor do we wish to prescribe, like Euclid, a definitive version of the elements of the subject. We are not committed to any particular philosophical programme. Nevertheless we have tried by critical discussion to ensure that each book represents a coherent line of thought; and that, by developing certain themes, it will be of greater interest than a mere assemblage of results and techniques. The books in the series differ in level: some are introductory some highly specialised. They also differ in scope: some offer a wide view of an area, others present a single line of thought. Each book is, at its own level, reasonably self-contained. Although no book depends on another as prerequisite, we have encouraged authors to fit their book in with other planned volumes, sometimes deliberately seeking coverage of the same material from different points of view. We have tried to attain a reasonable degree of uniformity of notation and arrangement. However, the books in the series are written by individual authors, not by the group. Plans for books are discussed and argued about at length. Later, encouragement is given and revisions suggested. But it is the authors who do the work; if, as we hope, the series proves of value, the credit will be theirs. History of the fi-Group. During 1968 the idea of an integrated series of monographs on mathematical logic was first mooted. Various discussions led to a meeting at Oberwolfach in the spring of 1969. Here the founding members of the group (R. 0. Gandy, A. Levy, G. H. Muller, G. Sacks, D. S. Scott) discussed the project in earnest and decided to go ahead with it. Professor F. K. Schmidt and Professor Hans Hermes gave us encouragement and support. Later Hans Hermes joined the group. To begin with all was fluid. How ambitious should we be? Should we write the books ourselves? How long would it take? Plans for authorless books were promoted, savaged and scrapped. Gradually there emerged a form and a method. At the end of an infinite discussion we found our name, and that of

VIII

Preface to the Series

the series. We established our centre in Heidelberg. We agreed to meet twice a year together with authors, consultants and assistants, generally in Oberwolfach. We soon found the value of collaboration: on the one hand the permanence of the founding group gave coherence to the over-all plans; on the other hand the stimulus of new contributors kept the project alive and flexible. Above all, we found how intensive discussion could modify the authors ideas and our own. Often the battle ended with a detailed plan for a better book which the author was keen to write and which would indeed contribute a perspective. Acknowledgements. The confidence and support of Professor Martin Earner of the Mathematisches Forschungsinstitut at Oberwolfach and of Dr. Klaus Peters ofSpringer- Verlag made possible thefirstmeeting and the preparation of a provisional plan. Encouraged by the Deutsche Forschungsgemeinschaft and the Heidelberger Akademie der Wissenschaften we submitted this plan to the Stiftung Volkswagenwerk where Dipl. Ing. Penschuck vetted our proposal; after careful investigation he became our adviser and advocate. We thank the Stiftung Volkswagenwerk for a generous grant (1970-73) which made our existence and our meetings possible. Since 1974 the work of the group has been supported by funds from the Heidelberg Academy; this was made possible by a special grant from the Kultusministerium von Baden-Wurttemberg (where Regierungsdirektor R. Goll was our counsellor). The success of the negotiations for this was largely due to the enthusiastic support of the former President of the Academy, Professor Wilhelm Doerr. We thank all those concerned. Finally we thank the Oberwolfach Institute, which provides just the right atmosphere for our meetings, Drs. Ulrich Feigner and Klaus Gloede for all their help, and our indefatigable secretary Elfriede Ihrig. Oberwolfach September 1975

R. O. Gandy A. Levy G. Sacks

H. Hermes G. H. Mutter D. S. Scott

Author's Preface

It is only before or after a book is written that it makes sense to talk about the reason for writing it. In between, reasons are as numerous as the days. Looking back, though, I can see some motives that remained more or less constant in the writing of this book and that may not be completely obvious. I wanted to write a book that would fill what I see as an artificial gap between model theory and recursion theory. I wanted to write a companion volume to books by two friends, H. J. Keisler's Model Theory for Infinitary Logic and Y.N. Moschovakis' Elementary Induction on Abstract Structures, without assuming material from either. I wanted to set forth the basic facts about admissible sets and admissible ordinals in a way that would, at long last, make them available to the logic student and specialist alike. I am convinced that the tools provided by admissible sets have an important role to play in the future of mathematical logic in general and definability theory in particular. This book contains much of what I wish every logician knew about admissible sets. It also contains some material that every logician ought to know about admissible sets. Several courses have grown out of my desire to write this book. I thank the students of these courses for their interest, suggestions and corrections. A rough first draft was written at Stanford during the unforgettable winter and spring of 1973. The book was completed at Heatherton, Freeland, Oxfordshire during the academic year 1973—74 while I held a research grant from the University of Wisconsin and an SRC Fellowship at Oxford. I wish to thank colleagues at these three institutions who helped to make it possible for me to write this book, particularly Professors Feferman, Gandy, Keisler and Scott. I also appreciate the continued interest expressed in these topics over the past years by Professor G. Kreisel, and the support of the Q-Group during the preparation of this book. I would like to thank Martha Kirtley and Judy Brickner for typing and John Schlipf, Matt Kaufmann and Azriel Levy for valuable comments on an earlier version of the manuscript. I owe a lot to Dana Scott for hours spent helping prepare the final manuscript. I would also like to thank Mrs. Nora Day and the other residents of Freeland for making our visit in England such a pleasant one.

X

Author's Preface

A final but large measure of thanks goes to my family: to Melanie for allowing me to use her room as a study during the coal strike; to Jon Russell for help with the corrections; but most of all to Mary Ellen for her encouragement and patience. To Mary Ellen, on this our eleventh anniversary, I promise to write at most one book every eleven years.

September 19, 1975 Santa Monica

K.J.B.

Table of Contents

Introduction

1

Part A. The Basic Theory

5

Chapter I. Admissible Set Theory

7

1. 2. 3. 4. 5. 6. 7. 8. 9.

The Role of Urelements The Axioms of KPU Elementary Parts of Set Theory in KPU Some Derivable Forms of Separation and Replacement Adding Defined Symbols to KPU Definition by Z Recursion The Collapsing Lemma Persistent and Absolute Predicates Additional Axioms

Chapter II. Some Admissible Sets 1. 2. 3. 4. 5. 6. 7. 8. 9.

The Definition of Admissible Set and Admissible Ordinal Hereditarily Finite Sets Sets of Hereditary Cardinality Less Than a Cardinal K Inner Models: the Method of Interpretations Constructible Sets with Urelements; HYP^ Defined Operations for Generating the Constructible Sets First Order Definability and Substitutable Functions The Truncation Lemma The L6vy Absoluteness Principle

Chapter HI. Countable Fragments of Lo^ 1. 2. 3. 4. 5.

Formalizing Syntax and Semantics in KPU Consistency Properties 9W-Logic and the Omitting Types Theorem A Weak Completeness Theorem for Countable Fragments Completeness and Compactness for Countable Admissible Fragments

7 9 11 14 18 24 30 33 38 42 42 46 52 54 57 62 69 72 76 78 78 84 87 92 95

XII

Table of Contents

6. The Interpolation Theorem 7. Definable Well-Orderings 8. Another Look at Consistency Properties Chapter IV. Elementary Results on HYP^ 1. 2. 3. 4. 5. 6. 7.

On Set Existence Defining n j and 1} Predicates IT} and A} on Countable Structures Perfect Set Results Recursively Saturated Structures Countable Wl-Admissible Ordinals Representability in 5R-Logic

103 105 109 113 113 116 122 127 137 144 146

PartB. The Absolute Theory

151

Chapter V. The Recursion Theory of 5^ Predicates on Admissible Sets . . 1. Satisfaction and Parametrization 2. The Second Recursion Theorem for KPU 3. Recursion Along Well-founded Relations 4. Recursively Listed Admissible Sets 5. Notation Systems and Projections of Recursion Theory 6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals 7. Ordinal Recursion Theory: Stability 8. Shoenfield's Absoluteness Lemma and the First Stable Ordinal . .

153 153 156 158 164 168

Chapter VI. Inductive Definitions

197

1. 2. 3. 4. 5. 6.

Inductive Definitions as Monotonic Operators Z Inductive Definitions on Admissible Sets First Order Positive Inductive Definitions and HYPan Coding H F ^ on 2R Inductive Relations on Structures with Pairing Recursive Open Games

173 177 189

197 205 211 220 230 242

Part C. Towards a General Theory

255

Chapter VII. More about L ^

257

1. 2. 3. 4. 5. 6. 7.

Some Definitions and Examples A Weak Completeness Theorem for Arbitrary Fragments Pinning Down Ordinals: the General Case Indiscernibles and upward Lowenheim-Skolem Theorems Partially Isomorphic Structures Scott Sentences and their Approximations Scott Sentences and Admissible Sets

257 262 270 276 292 297 303

Table of Contents

Chapter VIII. Strict 11} Predicates and Konig Principles 1. 2. 3. 4. 5. 6. 7. 8. 9.

The Konig Infinity Lemma Strict n{ predicates: Preliminaries Konig Principles on Countable Admissible Sets Konig Principles Kx and K2 on Arbitrary Admissible Sets Konig's Lemma and Nerode's Theorem: a Digression Implicit Ordinals on Arbitrary Admissible Sets Trees and Zx Compact Sets of Cofinality co Zx Compact Sets of Cofinality Greater than co Weakly Compact Cardinals

XIII

311 311 315 321 326 334 339 343 352 356

Appendix. Nonstandard Compactness Arguments and the Admissible Cover. 1. Compactness Arguments over Standard Models of Set Theory . . . 2. The Admissible Cover and its Properties 3. An Interpretation of KPU in KP 4. Compactness Arguments over Nonstandard Models of Set Theory .

365 365 366 372 378

References

380

Index of Notation

386

Subject Index

388

Major Dependencies

Appendix

(Va denotes the first three §§ of Chapter V, similarly for Vila.)

Introduction

Since its beginnings in the early sixties, admissible set theory has become a major source of interaction between model theory, resursion theory and set theory. In fact, for the student of admissible sets the old boundaries between fields disappear as notions merge, techniques complement one another, analogies become equivalences, and results in one field lead to results in another. This is the view of admissible sets we hope to share with the reader of this book. Model theory, recursion theory and set theory all deal, in part, with problems of definability and set existence. Definability theory is (by definition) that part of mathematical logic which deals with such problems. The Craig Interpolation Theorem, Kleene's analysis of Aj sets by means of the hyperarithmetic sets, Godel's universe L of constructible sets and Shoenfield's Absoluteness Lemma are all major contributions to definability theory. The theory of admissible sets takes such apparently divergent results and makes them converge in a single coherent body of thought, one with ramifications for all parts of logic. This book is written for the student who has taken a good first space year graduate course in logic. The specific material we presuppose can be summarized as follows. The student should understand the completeness, compactness and Lowenheim-Skolem theorems as well as the notion of elementary submodel. He should be familiar with the basic properties of recursive functions and recursively enumerable (hereinafter r.e.) sets. The student should have seen the development of intuitive set theory in some formal theory like ZF (ZermeloFraenkel set theory). His life will be more pleasant if he has some familiarity with the constructible sets before reading §§ II.5,6 or V.4—8, but our treatment of constructible sets is self-contained. A logical presentation of a reasonably advanced part of mathematics (which this book attempts to be) bears little relation to the historical development of that subject. This is particularly true of the theory of admissible sets with its complicated and rather sensitive history. On the other hand, a student is handicapped if he has no idea of the forces that figured in the development of his subject. Since the history of admissible sets is impossible to present here, we compromise by discussing how some of the older material fits into the current theory. We concentrate on those topics that are particularly relevant to this book. The prerequisites for understanding the introduction are rather greater than those for understanding the book itself.

2

Introduction

Recursive ordinals and hyper arithmetic sets. In retrospect, the study of admissible ordinals began with the work of Church and Kleene on notation systems and recursive ordinals (Church-Kleene [1937], Church [1938], Kleene [1938].) This study began as a recursive counterpart to the classical theory of ordinals; the least nonrecursive ordinal (o[ is the recursive analogue of col9 the first uncountable ordinal. (Similarly for coc2 and co2, etc.) The theory of recursive ordinals had its most important application when Kleene [1955] used it in his study of the class of hyperarithmetic sets, the smallest reasonably closed class of sets of natural numbers which can be considered as given by the structure Jf = (a>, + , x > of natural numbers. Kleene's theorem that hyperarithmetic = A} provided a construction process for the class of A} sets and constituted the first real breakthrough into (applied) second order logic. One of our aims is to provide a similar analysis for any structure $R. Given 5R we construct the smallest admissible set HYP W above $R (in § II.5) and use it in the study of definability problems over 9R (in Chapters IV and VI). The study of hyperarithmetic sets generated a lot of discussion of the analogy between, on the one hand, the 11} and hyperarithmetic sets, and the r.e. and recursive sets on the other. These analogies became particularly striking when expressed in terms of representability in co-logic and first order logic, by Grzegorczyk, Mostowski and Ryll-Nardzewski [1959]. The analogy had some defects, though, as the workers realized at the time. For example, the image of a hyperarithmetic function is hyperarithmetic, not just II} as the analogy would suggest. Kreisel [1961] analyzed this situation and discovered that the correct analogy is between 11} and hyperarithmetic on the one hand and r.e. and finite (not recursive) on the other. He went on to develop a recursion theory on the hyperarithmetic sets via a notation system. (He also proved the Kreisel Compactness Theorem for co-logic: If a 11} theory T of second order arithmetic is inconsistent in co-logic, then some hyperarithmetic subset T0^T is inconsistent in co-logic.) This theory was expanded in the metarecursion theory of KreiselSacks [1965]. Here one sees how to develop, by means of an ordinal notation system, an attractive recursion theory on CD\ such that for X^co: X is n } iff X is cocrr.e, X is A} iff X is co[-finite. In § IV.3 we generalize this, by means of HYP^, to show that for any countable structure 9W and any relation R on $R: R is n } on m iff R is HYP^-r.e., R is A} on Wl iff R is HYP^-finite, thus providing a construction process for the A} relations over any countable structure SR whatsoever. The use of notation systems then allows us to transfer results from HYP^ to SR itself (see §§ V.5 and VI.5).

Introduction

3

Constructible sets. The other single most important line of development leading to admissible sets also goes back to the late thirties. It began with the introduction by Godel [1939] of the class L of constructible sets, in order to provide a model of set theory satisfying the axiom of choice and generalized continuum hypothesis (GCH). Takeuti [1960, 1961] discovered that one could develop L by means of a recursion theory on the class Ord of all ordinals. He showed that Godel's proof of the GCH in L corresponds to the following recursion theoretic stability: If K is an uncountable cardinal and if F: Ord -• Ord is ordinal-recursive then F(P)0} = the torsion subgroup of G, D = \J {H | H is a divisible subgroup of G} = the divisible part of G.

2. The Axioms of KPU

9

While these definitions are clearly increasing in logical complexity, there is no distinction to be made between them from ZF's point of view. We will return to this example in Chapter IV. 1.3. As an example of the way one is tempted to violate the principle of parsimony when working in ZF, one need only look in the average text on set theory. There you will find the power set axiom (a very strong axiom from our point of view) used to verify a simple fact like the existence of axb. 1.4. The point made in (4) above is illustrated by considering the real line. While we know how to construct something isomorphic to the real line in ZF (either by Cauchy sequences or by Dedekind cuts), in practise the mathematician is not interested in the details of this construction. For example, he would never think of worrying about what the elements of ]/2 happen to be. 1.5 Notes. The notes at the end of sections are used to collect historical remarks, credit for theorems (when possible) and various remarks which might otherwise have gone into footnotes. In the early days of set theory, certainly in the work of Zermelo, urelements were an integral part of the subject. The rehabilitation of urelements in the context of admissible set theory is such a simple idea that it would be silly to assign credit for it to any one person. Probably everyone who has thought at all about infinitary logic and admissible sets has had a similar idea. Karp [1968] suggests the study of nontransitive admissible sets. Kreisel [1971] points out that "the principal gap in the existing model theoretic [generalized recursion theory] ... is its preoccupation with sets (that is sets built up from the empty set by some cumulative operation...); not even sets of individuals are treated." Barwise [1974] contains the first published treatment of admissible sets with urelements. This book grew out of that paper, to some extent. It is worth remembering that the defense of urelements given in § 1 would have been unnecessary not too long ago. Perhaps it will be equally pointless sometime in the future.

2. The Axioms of KPU Let L be a first order language with equality, some relation, function and constant symbols and let SER = be a structure for this language L We wish to form admissible sets which have M as a collection of urelements; these admissible sets are the intended models of a theory KPU which we begin to develop in this section. The theory KPU is formulated in a language L* = L(e,...) which extends L by adding a membership symbol e and, possibly other function, relation and constant symbols. Rather than describe L* precisely, we describe its class of structures, leaving it to the reader to formalize L* in a way that suits his tastes.

10

I. Admissible Set Theory

2.1 Definition. A structure 91^ = (9K; A,£,...) for L* consists of (i) a structure 9K = for the language L, where M = 0 is kept open as a possibility (the members of M are the urelements of 91^); (ii) a nonempty set ,4 disjoint from M (the members of A are the sets of 91^); (iii) a relation £ g ( M u i ) x i (which interprets the membership symbol e); (iv) other functions, relations and constants on M u A to interpret any other symbols in L(e,...) (that is the symbols in the list indicated by the three dots). The equality symbol of L* is always interpreted as the usual equality relation. We use variables of L* subject to the following conventions: Given a structure 2IOT = (aR;4,£,...) for L*, p,g,p 1? ...

range over M (urelements),

a,b,c,d,/,r,a l5 ... range over A (sets), x,y,z,...

range over M u i

This notation gives us an easy way to assert that something holds of sets, or of urelements. For example, Vp3aVx (xeaq = p) asserts that there is a set a whose intersection with the class of all urelements is {p}. We sometimes use (e.g. in 2.2(iii)) u,v,w to denote any kind of variable. The axioms of KPU are of three kinds. The axioms of extensionality and foundation concern the basic nature of sets. The axioms of pair, union and Ao separation deal with the principles of set construction available to us. The most important axiom, Ao collection, guarantees that there are enough stages in our construction process. In order to state the latter two axioms we need to define the notion of Ao formula of L(e,...), of Levy [1965]. 2.2 Definition. The collection of Ao formulas of a language L(e,...) is the smallest collection Y containing the atomic formulas of L(e,...) closed under: (i) if cp is in Y, then so is —\q>\ (ii) if (p,ij/ are in Y, so are (q> A ij/) and ((pvij/); (iii) if cp is in Y, then so are \fuevcp and luevcp for all variables u and v. The importance of Ao formulas rests in the metamathematical fact that any predicate defined by a Ao formula is absolute (see 7.3), and the empirical fact (which we will verify) that many predicates occuring in nature can be defined by Ao formulas (see Table 1). 2.3 Definition. The theory KPU (relative to a language L(e,...)) consists of the universal closures of the following formulas: Extensionality: Vx (xea =