CUUS1074-Lerman
March 23, 2010
11:41
978 0 521 76065 2
This page intentionally left blank
ii
Logic Colloquium 20...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

CUUS1074-Lerman

March 23, 2010

11:41

978 0 521 76065 2

This page intentionally left blank

ii

Logic Colloquium 2007 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 2007, 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. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay’s conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

Fran¸coise Delon was Directrice d’´etudes at the Centre de Formation des PEGC of Reims and Humboldt Stipendiatin at Freiburg and is presently a Directrice de Recherche at Centre National de la Recherche Scientifique. Ulrich Kohlenbach is a Professor of Mathematics at TU Darmstadt (Germany). He is the coordinating editor of Annals of Pure and Applied Logic and the president of the Deutsche Vereinigung f¨ur Mathematische Logik und f¨ur Grundlagen der Exakten Wissenschaften. Penelope Maddy is a Distinguished Professor of Logic and Philosophy of Science at the University of California, Irvine. She is a Fellow of the American Academy of Arts and Sciences and is currently the president of the Association for Symbolic Logic. Frank Stephan is an Associate Professor in the departments of mathematics and computer science at the National University of Singapore. He is the editor of the Journal of Symbolic Logic.

CUUS1074-Lerman

March 23, 2010

11:41

978 0 521 76065 2

ii

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 dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board H. Dugald Macpherson, Managing Editor Department of Pure Mathematics, School of Mathematics, University of Leeds Jeremy Avigad Department of Philosophy, Carnegie Mellon University Vladimir Kanovei Institute for Information Transmission Problems, Moscow Manuel Lerman Department of Mathematics, University of Connecticut Heinrich Wansing Institute of Philosophy, Dresden University of Technology Thomas Wilke Institut f¨ur Informatik, Christian-Albrects-Universitat zu Kiel

More information, including a list of the books in the series, can be found at http://www.aslonline.org/books-lnl.html.

Logic Colloquium 2007 Edited by

FRANC¸ OISE DELON UFR de Math´ematiques

ULRICH KOHLENBACH Technische Universit¨at Darmstadt

PENELOPE MADDY University of California, Irvine

FRANK STEPHAN National University of Singapore

association for symbolic logic

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/9780521760652 © Association of Symbolic Logic 2010 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 2010 ISBN-13

978-0-511-78785-0

eBook (EBL)

ISBN-13

978-0-521-76065-2

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.

CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Speakers and Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

ˇ ci´c, Pavel Pudl´ak, Greg Restall, Alasdair Urquhart, and Vedran Caˇ Albert Visser Decorated linear order types and the theory of concatenation . . . . . . .

1

Andr´es Eduardo Caicedo Cardinal preserving elementary embeddings . . . . . . . . . . . . . . . . . . . . . . . .

14

Fernando Ferreira Proof interpretations and majorizability . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Philipp Gerhardy Proof mining in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Steve Jackson Cardinal structure under AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Bakhadyr Khoussainov and Mia Minnes Three lectures on automatic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Ya’acov Peterzil Pillay’s conjecture and its solution—a survey . . . . . . . . . . . . . . . . . . . . . . . 177 Greg Restall Proof theory and meaning: On the context of deducibility . . . . . . . . . . 204 Marcus Tressl Bounded super real closed rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Andreas Weiermann Analytic combinatorics of the transﬁnite: A unifying Tauberian perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

vii

INTRODUCTION

The Logic Colloquium 2007, the European Summer Meeting of the Association for Symbolic Logic, was held in Wrocław, Poland, from 14 to 19 July 2007. It was colocated with the following events: the thirty-fourth International Colloquium on Automata, Languages and Programming (ICALP); the twenty-second Annual IEEE Symposium on Logic in Computer Science (LICS) and the ninth ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP). There was an agreement with LICS on running joint sessions for one day. More than 200 participants from all over the world took part in the Logic Colloquium. The programme consisted of 3 tutorials, 11 invited plenary talks, 6 joint talks with LICS (2 long, 4 short) and 21 talks in 5 special sessions on set theory, proof complexity and nonclassical logics, philosophical and applied logic at the JPL, logic and analysis and model theory. In addition to these invited talks, there were 63 contributed talks. The programme committee consisted of Alessandro Andretta (Turin), Franc¸oise Delon (Paris 7), Ulrich Kohlenbach (Darmstadt), Steﬀen Lempp (Madison, Chair), Penelope Maddy (UC Irvine), Jerzy Marcinkowski (Wrocław), Ludomir Newelski (Warsaw), Andrew Pitts (Cambridge), Pavel Pudl´ak (Prague), Sławomir Solecki (Urbana-Champaign), Frank Stephan ¨ (Singapore) and Goran Sundholm (Leiden). The local organizing committee consisted of Tobias Kaiser, Piotr Kowalski, Jan Kraszewski, Amador MartinPizarro, Serge Randriambololona and Roman Wencel. The Logic Colloquium 2007 wants to acknowledge its sponsors for their generous support of the event: the Association for Symbolic Logic, the Polish Academy of Sciences and the University of Wrocław. The next pages give an overview of the programme of the meeting; in addition to the talks listed, there were contributed talks from all ﬁelds of mathematical logic. Speakers invited to give a plenary or special session talk were also ix

x

INTRODUCTION

invited to contribute to this volume. The contributions handed in were refereed according to the high standards of logic journals. We would like to thank the authors for their excellent contributions as well as the referees for their diligent work to review and evaluate the submissions. The Editors Franc¸oise Delon Ulrich Kohlenbach Penelope Maddy Frank Stephan

SPEAKERS AND TITLES

Tutorials Steve Jackson (North Texas): Cardinal arithmetic in L(R). Bakh Khoussainov (Auckland): Automatic structures. Kobi Peterzil (Haifa): The inﬁnitesimal subgroup of a deﬁnably compact group.

Invited Plenary Talks Albert Atserias (Barcelona): Structured ﬁnite model theory. Matthias Baaz (Vienna): Towards a proof theory of analogical reasoning. Vasco Brattka (Cape Town): Computable analysis and eﬀective descriptive set theory. Zo´e Chatzidakis (Paris 7): Model theory of diﬀerence ﬁelds and some applications. Gabriel Debs (Paris 6): Coding compact spaces of Borel functions. Fernando Ferreira (Lisbon): On a new functional interpretation. Andrzej Grzegorczyk (Warsaw): Philosophical content of formal achievements. Bjørn Kjos-Hanssen (Hawaii): Brownian motion and Kolmogorov complexity. Piotr Kowalski (Wrocław): Deﬁnability in diﬀerential ﬁelds. Paul Larson (Miami U): Large cardinals and forcing-absoluteness. Tony Martin (UC Los Angeles): Sets and the concept of set. xi

xii

SPEAKERS AND TITLES

LICS Joint Long Talks Martin Hyland (Cambridge): Combinatorics and proofs. Colin Stirling (Edinburgh): Higher-order matching, games and automata.

LICS Joint Short Talks Cristiano Calcagno (Imperial College): Can logic tame systems programs? Mart´ın Escardo´ (Birmingham): Inﬁnite sets that admit exhaustive search. Rosalie Iemhoﬀ (Utrecht): Skolemization in constructive theories. Alex Simpson (Edinburgh): Non-well-founded proofs.

Special Session Set Theory Organized by Ilijas Farah and Joel Hamkins. M´arton Elekes (Hungarian Academy of Sciences): Partitioning κ-fold covers into κ many subcovers. ¨ Gunter Fuchs (Munster): Maximality principles for closed forcings. Victoria Gitman (City University of New York): Scott’s problem for proper Scott sets. Lionel Nguyen Van The (Calgary): The Urysohn sphere is oscillation stable. Asger Tornquist (Toronto): Classifying measure preserving actions up to conjugacy and orbit equivalence. Matteo Viale (Vienna): The constructible universe for the anti-foundation axiom system ZFA.

Special Session Proof Complexity and Nonclassical Logics Organized by Pavel Pudl´ak. Emil Jer´abek (Prague): Proof systems for modal logics. George Metcalfe (Vanderbilt): Substructural fuzzy logics. Alasdair Urquhart (Toronto): Complexity problems for substructural logics.

SPEAKERS AND TITLES

xiii

Special Session Philosophical and Applied Logic at the JPL Organized by Penelope Maddy. Aldo Antonelli (UC Davis): From philosophical logic to computer science. Horacio Arlo` Costa (Carnegie-Mellon): Philosophical logic meets formal epistemology. Greg Restall (Melbourne): Proof theory and meaning: three case studies. Albert Visser (Utrecht): Interpretations in philosophical logic.

Special Session Logic and Analysis Organized by Ita¨ı Ben Yaacov and Ulrich Kohlenbach. Philipp Gerhardy (Pittsburgh/Oslo): Local stability of ergodic averages. ¨ Peter Hertling (Univ. der Bundeswehr Munchen): Computability and non-computability results for the topological entropy of shift spaces. Julien Melleray (Urbana-Champaign): Geometry of the Urysohn space: a model-theoretic approach. Andreas Weiermann (Ghent): Analytic combinatorics of the transﬁnite.

Special Session Model Theory Organized by Franc¸oise Delon and Ludomir Newelski. Vera Djordjevic (Uppsala): Independence in structures and ﬁnite satisﬁability. Amador Martin-Pizarro (Lyon 1): Some thoughts on bad objects. Ziv Shami (Tel Aviv): Countable imaginary simple unidimensional theories. Marcus Tressl (Regensburg): Super real closed rings.

DECORATED LINEAR ORDER TYPES AND THE THEORY OF CONCATENATION

ˇ CI ˇ C, ´ PAVEL PUDLAK, ´ VEDRAN CA GREG RESTALL, ALASDAIR URQUHART, AND ALBERT VISSER

Abstract. We study the interpretation of Grzegorczyk’s Theory of Concatenation TC in structures of decorated linear order types satisfying Grzegorczyk’s axioms. We show that TC is incomplete for this interpretation. What is more, the ﬁrst order theory validated by this interpretation interprets arithmetical truth. We also show that every extension of TC has a model that is not isomorphic to a structure of decorated order types. We provide a positive result, to wit a construction that builds structures of decorated order types from models of a suitable concatenation theory. This construction has the property that if there is a representation of a certain kind, then the construction provides a representation of that kind.

§1. Introduction. In his paper [2], Andrzej Grzegorczyk introduces a theory of concatenation TC. The theory has a binary function symbol ∗ for concatenation and two constants a and b. The theory is axiomatized as follows. TC1. (x ∗ y) ∗ z = x ∗ (y ∗ z) TC2. x ∗ y = u ∗ v → ((x = u ∧ y = v) ∨ ∃w ((x∗w = u∧y = w∗v)∨(x = u∗w∧y∗w = v))) TC3. x ∗ y = a TC4. x ∗ y = b TC5. a = b Axioms TC1 and TC2 are due to Tarski [7]. Grzegorczyk calls axiom TC2 the editor axiom. We will consider two weaker theories. The theory TC0 has the signature with just concatenation, and is axiomatized by TC1,2. The theory TC1 is axiomatized by TC1,2,3. We will also use TC2 for TC. ´ ¸ z` a of the Some of the results of this note were obtained during the Excursion to mountain Sle inspiring Logic Colloquium 2007 in Wrocław and, in part, in the evening after the Excursion. We thank the organizers for providing this wonderful opportunity. We thank Dana Scott for his comments, insights and questions. We are grateful to Vincent van Oostrom for some perceptive remarks. Pavel Pudl´ak was supported by grants A1019401 and 1M002162080. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

1

2

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

The theories we are considering have various interesting interpretations. First they are, of course, theories of strings with concatenation; in other words, theories of free semigroups. Secondly they are theories of wider classes of structures, to wit structures of decorated linear order types, which will be deﬁned below1 . The theories TCi are theories for concatenation without the empty string, i.e., without the unit element. Adding a unit ε one obtains another class of theories TCεi , theories of free monoids, or theories of structures of decorated linear order types including the empty linear decorated order type. The basic list of axioms is as follows. TCε 1. ε ∗ x = x ∧ x ∗ ε = x TCε 2. (x ∗ y) ∗ z = x ∗ (y ∗ z) TCε 3. x ∗y = u ∗v → ∃w ((x ∗w = u ∧y = w ∗v)∨(x = u ∗w ∧w ∗y = v)) TCε 4. a = ε TCε 5. x ∗ y = a → (x = ε ∨ y = ε) TCε 6. b = ε TCε 7. x ∗ y = b → (x = ε ∨ y = ε) TCε 8. a = b. We take TCε0 to be the theory axiomatized by TCε 1, 2, 3, TCε1 to be TCε0 + TCε 4, 5 and TCε := TCε2 to be TCε1 + TCε 6, 7, 8. One can show that TC is bi-interpretable with TCε , in which a unit ε is added via one dimensional interpretations without parameters. The theory TC1 is biinterpretable with TCε1 via two-dimensional interpretations with parameters. The situation for TC0 seems to be more subtle. See also [10]. In Section 6, we will study an extension of TCε0 . Andrzej Grzegorczyk and Konrad Zdanowski have shown that TC is essentially undecidable. This result can be strengthened by showing that Robinson’s Arithmetic Q is mutually interpretable with TC. Note that TC0 is undecidable —since it has an extension that parametrically interprets TC— but that TC0 is not essentially undecidable: it is satisﬁed by a one-point model. Similarly TC1 is undecidable, but it has as an extension the theory of ﬁnite strings of a’s, which is a notational variant of Presburger Arithmetic and, hence, decidable. We will call models of TC0 concatenation structures, and we will call models of TCi concatenation i-structures. The relation of isomorphism between concatenation structures will be denoted by ∼ =. We will be interested in concatenation structures, whose elements are decorated linear order types with the operation concatenation of decorated order types. Let a non-empty class A be given. An A-decorated linear ordering is a structure D, ≤, f , where D is a non-empty domain, ≤ is a linear ordering on D, and f is a function from 1 A special case of decorated linear order types is addition of sets as discovered by Tarski (see [8]). It is shown by Laurence Kirby in [3] that the structure of addition on sets is isomorphic to addition of well-founded order types with a proper class of decorating objects.

DECORATED LINEAR ORDER TYPES

3

D to A. A mapping φ is an isomorphism between A-decorated linear order types D, ≤, f and D , ≤ , f iﬀ it is a bijection between D and D such that, for all d, e in D, d ≤ e ⇔ φd ≤ φe, and fd = f φd . Our notion of isomorphism gives us a notion of A-decorated linear order type. We have an obvious notion of concatenation between A-decorated linear orderings which induces a corresponding notion of concatenation for A-decorated linear order types. We use α, , . . . to range over such linear order types. Since, linear order types are classes we have to follow one of two strategies: either to employ Scott’s trick to associate a set object to any decorated linear order type or to simply refrain from dividing out isomorphism but to think about decorated linear orderings modulo isomorphism. We will employ the second strategy. We will call a concatenation structure whose domain consists of (representatives of) A-decorated order types, for some A, and whose concatenation is concatenation of decorated order types: a concrete concatenation structure. It seems entirely reasonable to stipulate that e.g. the interpretation of a in a concrete concatenation structure is a decorated linear order type of a one element order. However, for the sake of generality we will refrain from making this stipulation. Grzegorczyk conjectured that every concatenation 2-structure is isomorphic to a concrete concatenation structure. We prove that this conjecture is false. (i) Every extension of TC1 has a model that is not isomorphic to a concrete concatenation 1-structure and (ii) the set of principles valid in all concrete concatenation 2-structures interprets arithmetical truth. The plan of the paper is as follows. We show, in Section 2, that we have, for all decorated order types α, and , the following principle: (†)

∗ α ∗ = α ⇒ ∗ α = α ∗ = α.

This fact was already known. It is due to Lindenbaum, credited to him in ´ Sierpinski’s book [6] on p. 248. It is also problem 6.13 of [4]. It is easy to see that every group is a concatenation structure and that (†) does not hold in the two element group. We show, in Section 5, that every concatenation structure can be extended to a concatenation structure with any number of atoms. It follows that there is a concatenation structure with at least two atoms in which (†) fails. Hence, TC is incomplete for concrete concatenation structures. In Section 3, we provide a counterargument of a diﬀerent ﬂavour. We provide a tally interpretation that deﬁnes the natural numbers (with concatenation in the role of addition) in every concrete concatenation 2structure. It follows that every extension of TC1 is satisﬁed by a concatenation 1-structure that is not isomorphic to any concrete concatenation 1-structure, to wit any model of that extension that contains a non-standard element. In Section 4, we strengthen the result of Section 3, by showing that in concrete concatenation 2-structures we can add multiplication to the natural numbers.

4

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

It follows that the set of arithmetically true sentences is interpretable in the concretely valid consequences of TC2 . Finally, in Section 6 we prove a positive result. We provide a mapping from arbitrary models of a variant of an extension of TC0 to structures of decorated order types. As we have shown such a construction cannot always provide a representation. We show that, for a restricted class of representations, we do have: if a model has a representation in the class, then the construction yields such a representation. §2. A principle for decorated order types. In this section we prove a universal principle that holds in all concatenation structures, which is not provable in TC. There is an earlier proof of this principle [4, p. 187]. Our proof, however, is diﬀerent from that of Komj´ath and Totik. Theorem 1. Let α0 , α1 , α2 be decorated order types. Suppose that α1 = α0 ∗ α1 ∗ α2 . Then, α1 = α0 ∗ α1 = α1 ∗ α2 . Proof. Suppose α1 = α0 ∗ α1 ∗ α2 . Consider a decorated linear ordering A := A, ≤, f of type α1 , By our assumption, we may partition A into A0 , A1 , A2 , such that:

A, ≤, f = A0 , ≤ A0 , f A0 ∗ A1 , ≤ A1 , f A1 ∗ A2 , ≤ A2 , f A2 , where Ai := Ai , ≤ Ai , f Ai is an instance of αi , Let φ : A → A1 be an isomorphism. Let φ n A(i) := φ n [A(i) ], ≤ φ n [A(i) ], f φ n [A(i) ] . We have: φ n Ai is of order type αi and φ n A is of order type α1 . Clearly, φA0 is an initial substructure of φA = A1 . So, A0 and φA0 are to the right of A0 . Similarly, for φ n A0 and φ n+1 A0 . disjoint and φA 0 adjacent i Take A0 := i∈ φ A0 . We ﬁnd that A 0 := A0 , ≤ A0 , f A0 is initial in A and of decorated linear order type α0 . So α1 = α0 ∗ , for some . It follows that α0 ∗ α1 = α0 ∗ α0 ∗ = α0 ∗ = α1 . The other identity is similar. So, every concrete concatenation structure validates that α1 = α0 ∗ α1 ∗ α2 implies α1 = α0 ∗ α1 = α1 ∗ α2 . We postpone the proof that this principle is not provable in TC to Section 5. §3. Deﬁnability of the natural numbers. In this section, we show that the natural numbers can be deﬁned in every concrete concatenation 1-structure. We deﬁne: • • • •

x ⊆ y :↔ x = y ∨ ∃u (u ∗ x = y)∨ ∃v (x ∗ v = y)∨ ∃u, v (u ∗ x ∗ v = y). x ⊆ini y :↔ x = y ∨ ∃v (x ∗ v = y). x ⊆end y :↔ x = y ∨ ∃u (u ∗ x = y). a ) :↔ ∀m⊆ini n (m = a ∨ ∃k (k = m ∧ m = k ∗ a)). (n : N

DECORATED LINEAR ORDER TYPES

5

a is derived from the analogous use in type theory. We The use of ‘:’ in n : N a for: (m : N a ) ∧ (n : N a ), could read it as: n is of sort Na . We write m, n : N a a with the extension of N etc. In the context of a structure we will confuse N in that structure. We prove the main theorem of this section. Theorem 2. In any concrete concatenation 1-structure, we have: a = an+1 | n ∈ . N a is precisely the class of natural numbers in tally representation In other words, N (starting with 1). Note that ∗ on this set is addition. Proof. Consider any concrete concatenation 1-structure A. It is easy to a . Clearly, every element x of N a is either a or it see that every an+1 is in N has a predecessor, i.e., there is a y such that x = y ∗ a. The axioms of TC1 guarantee that this predecessor is unique. This justiﬁes the introduction of the a . Let α be the order type corresponding partial predecessor function pd on N to a. Let 0 be any element of Na . If, for some n, pdn 0 is undeﬁned, then 0 is clearly of the form α k+1 , for k in . We show that the other possibility cannot obtain. Suppose n := pdn 0 is always deﬁned. Let A be a decorated linear ordering of type α and let Bi be a decorated linear ordering of type i . We assume that the domain A of A is disjoint from the domains Bi of the Bi . Thus, we may implement Bi+1 ∗ A just by taking the union of the domains. Let φi be isomorphisms from Bi+1 ∗ A to Bi . Let Ai := (φ0 ◦ · · · ◦ φi )(A). Then, the Ai are all of type α and, for some C, we have B0 ∼ = C ∗ · · · ∗ A1 ∗ A 0 . Similarly B1 ∼ = C ∗ · · · ∗ A2 ∗ A1 . Let ˘ be the opposite ordering of . a2. A It follows that 0 = ∗ α ˘ = 1 = pd(0 ). Hence, 0 is not in N contradiction. We call a concatenation structure standard if Na deﬁnes the tally natural numbers. Since, by the usual argument, any extension of TC1 has a model with non-standard numbers, we have the following corollary. Corollary 3. Every extension of TC1 has a model that is not isomorphic to a concrete concatenation 1-structure. In a diﬀerent formulation: for every concatenation 1-structure there is an elementarily equivalent concatenation 1structure that is not isomorphic to a concrete concatenation 1-structure. Note that the non-negative tally numbers with addition form a concrete concatenation 1-structure. Thus, the concretely valid consequences of TC1 + a ), i.e., the principles valid in every concrete concatenation 1∀x (x : N a ) are decidable. structure satisfying ∀x (x : N 2 Note

that we are not assuming that is in A.

6

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

§4. Deﬁnability of multiplication. If we have two atoms to work with, we can add multiplication to our tally numbers. This makes the set of concretely valid consequences of TC non-arithmetical. The main ingredient of the deﬁnition of multiplication is the theory of relations on tally numbers. In TC, we can develop such a theory. The development has some resemblance to the construction in the classic paper of Quine [5]. However, the ideas here are somewhat more intricate, since we areworking in a more generalcontext than that of [5]. We represent the relation x0 , y0 , . . . , xn−1 , yn−1 , by: bb ∗ x0 ∗ b ∗ y0 ∗ bb ∗ x1 ∗ . . . bb ∗ xn−1 ∗ b ∗ yn−1 ∗ bb. We deﬁne: • r : REL :↔ bb ⊆end r, • ∅ := bb, a ∧ bb ∗ x ∗ b ∗ y ∗ bb ⊆ r. • x[r]y :↔ x, y : N • adj(r, x, y) := r ∗ x ∗ b ∗ y ∗ bb. Clearly, we have: TC ∀u, v ¬ u[∅]v. To verify that this coding works we need the adjunction principle. Theorem 4. We have: a) → TC (r : REL ∧ x, y, u, v : N u[adj(r, x, y)]v ↔ u[r]v ∨ (u = x ∧ v = y) . We can prove this result by laborious and unperspicuous case splitting. However, it is more elegant to do the job with the help of a lemma. Consider any model of TC0 . Fix an element w. We call a sequence (w0 , . . . , wk ) a partition of w if we have that w0 ∗ · · · ∗ wk = w. The partitions of w form a category with the following morphisms. f : (u0 , . . . , un ) → (w0 , . . . , wk ) iﬀ f is a surjective and weakly monotonic function from n + 1 to k + 1, such that, for any i ≤ k, wi = us ∗ · · · ∗ u , where f(j) = i iﬀ s ≤ j ≤ . We write (u0 , . . . , un ) ≤ (w0 , . . . , wk ) for: ∃f f : (u0 , . . . , un ) → (w0 , . . . , wk ). In this case we say that (u0 , . . . , un ) is a reﬁnement of (w0 , . . . , wk ). Lemma 5. Consider any concatenation structure. Let w be an element of the structure. Then, any two partitions of w have a common reﬁnement. Proof. Fix any concatenation structure. We ﬁrst prove that, for all w, all pairs of partitions (u0 , . . . , un ) and (w0 , . . . , wk ) of w have a common reﬁnement, by induction of n + k. If either n or k is 0, this is trivial. Suppose that (u0 , . . . , un+1 ) and (w0 , . . . , wk+1 ) are partitions of w. By the editor axiom, we have either (a) u0 ∗ · · · ∗ un = w0 ∗ · · · ∗ wk and un+1 = wk+1 , or there is a v such that (b) u0 ∗ · · · ∗ un ∗ v = w0 ∗ · · · ∗ wk and un+1 = v ∗ wk+1 , or (c) u0 ∗ · · · ∗ un = w0 ∗ · · · ∗ wk ∗ v and v ∗ un+1 = wk+1 . We only treat case (b), the other cases being easier or similar. By the induction hypothesis, there is a common reﬁnement (x0 , . . . xm ) of (u0 , . . . , un , v) and (w0 , . . . , wn ). Let this

DECORATED LINEAR ORDER TYPES

7

be witnessed by f, resp. g. It is easily seen that (x0 , . . . xm , wk+1 ) is the desired reﬁnement with witnessing functions f and g , where f := f[m+1 → n+1], g := g[m + 1 → k + 1]. Here f[m + 1 → n + 1] in the result of extending f to assign n + 1 to m + 1. We turn to the proof of Theorem 4. The veriﬁcation proceeds more or less as one would do it for ﬁnite strings. Proof. Consider any concatenation 2-structure. Suppose REL(r). The right-to-left direction is easy, so we treat left-to-right. Suppose x, y, u and v are tally numbers. and u[adj(r, x, y)]v. There are two possibilities. Either r = bb or r = r0 ∗ bb. We will treat the second case. Let s := adj(r, x, y). One the following four partitions is a partition of s: (i) (b, b, u, b, v, b, b), or (ii) (w, b, b, u, b, v, b, b), or (iii) (b, b, u, b, v, b, b, z), or (iv) (w, b, b, u, b, v, b, b, z). We will treat cases (ii) and (iv). Suppose := (w, b, b, u, b, v, b, b) is a partition of s. We also have that := (r0 , b, b, x, b, y, b, b) is a partition of s. Let (t0 , . . . , tk ) be a common reﬁnement of and , with witnessing functions f and g. The displayed b’s in these partitions must have unique places among the ti . We deﬁne m to be the unique i such that f(i) = m, provided that m = b. Similarly, for m . (To make this unambiguous, we assume that if = , we take as the common reﬁnement with f and g both the identity function.) We evidently have 7 = 7 = k and 6 = 6 = k − 1. Suppose 4 < 4 . It follows that b ⊆ v. So, v would have an initial subsequence that ends in b, which is impossible. So, 4 < 4 . Similarly, 4 < 4 . So 4 = 4 . It follows that v = y. Reasoning as in the case of 4 and 4 , we can show that 2 = 2 and, hence u = x. Suppose := (w, b, b, u, b, v, b, b, z) is a partition of s. We also have that := (r0 , b, b, x, b, y, b, b) is a partition of s. Let (t0 , . . . , tk ) be a common reﬁnement of and , with witnessing functions f and g. We consider all cases, where 1 < 6 . Suppose 6 = 1 + 1 = 2 . Note that 7 = 6 + 1, so we a . Suppose 2 < 6 < 4 . In this case we ﬁnd: b ⊆ x, quod non, since x is in N have a b as substring of x. Quod non. Suppose 6 = 4 . Since 7 = 6 + 1, we get a b in y. Quod non. Suppose 4 < 6 < 6 . In this case, we get a b in y. Quod impossible. Suppose 6 ≥ 6 = k − 1. In this place there is no place left for z among the ti . So, in all cases, we obtain a contradiction. So the only possibility is 6 ≤ 1 . Thus, it follows that u[r]v. We can now use our relations to deﬁne multiplication of tally numbers in the usual way. See e.g. Section 2.2 of [1]. In any concrete concatenation 2-structure, we can use induction to verify the deﬁning properties of multiplication as deﬁned. It follows that we can interpret all arithmetical truths in the set of concretely valid consequences of TC. Corollary 6. We can interpret true arithmetic in the set of all principles valid in concrete concatenation 2-structures.

8

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

§5. The sum of concatenation structures. In this section we show that concatenation structures are closed under sums. This result will make it possible to verify the claim that the universal principle of Section 2 is not provable in TC. The result has some independent interest, since it provides a good closure property of concatenation structures. Consider two concatenation structures A0 and A1 . We write for concatenation in the Ai . We may assume, without loss of generality, that the domains of A0 and A1 are disjoint. We deﬁne the sum B := A0 ⊕ A1 as follows. • The domain of B consists of non-empty sequences w0 · · · wn−1 , where the wj are alternating between elements of the domains of A0 and A1 . In other words, if wj is in the domain of Ai , then wj+1 , if it exists, is in the domain of A1−i . • The concatenation ∗ of := w0 · · · wn−1 and := v0 · · · vk−1 is w0 · · · wn−1 v0 · · · wk−1 , in case wn−1 and v0 are in the domains of diﬀerent structures Ai . The concatenation ∗ is w0 · · · (wn−1 v0 ) · · · wk−1 , in case wn−1 and v0 are in in the same domain. In case ∗ is obtained via the ﬁrst case, we say that and are glued together. If the second case obtains, we say that and are clicked together. Theorem 7. The structure B = A0 ⊕ A1 is a concatenation structure. Proof. Associativity is easy. We check the editor property TC2. Suppose

0 ∗ 1 = z0 · · · zm−1 = 0 ∗ 1 . We distinguish a number of cases. Case 1. Suppose both of the pairs 0 , 1 and 0 , 1 are glued together. Then, for some k, n > 0, we have 0 = z0 · · · zk−1 , 1 = zk · · · zm−1 , 0 = z0 · · · zn−1 , and 1 = zn · · · zm−1 . So, if k = n, we have 0 = 0 and 1 = 1 . If k < n, we have 0 = 0 ∗ (zk · · · zn−1 ) and 1 = (zk · · · zn−1 ) ∗ 1 . The case that n < k is similar. Case 2. Suppose 0 , 1 is glued together and that 0 , 1 is clicked together. So, there are k, n > 0, u0 , and u1 such that 0 = z0 · · · zk−2 u0 , 1 = u1 zk · · · zm−1 , u0 u1 = zk−1 , 0 = z0 · · · zn−1 , and 1 = zn · · · zm−1 . Suppose k ≤ n. Then, 0 = 0 ∗(u1 zk · · · zn−1 ) and 1 = (u1 zk · · · zn−1 )∗1 . Note that, in case k = n, the sequence zk · · · zn−1 is empty. The case that k ≥ n is similar. Case 3. This case, where 0 , 1 is clicked together and 0 , 1 is glued together, is similar to Case 2. Case 4. Suppose that 0 , 1 and 0 , 1 are both clicked together. So, there are k, n > 0, u0 , u1 , v0 , v1 such that 0 = z0 · · · zk−2 u0 , 1 = u1 zk · · · zm−1 , u0 u1 = zk−1 , 0 = z0 · · · zn−2 v0 , 1 = v1 zn · · · zm−1 and v0 v1 = zn−1 .

DECORATED LINEAR ORDER TYPES

9

Suppose k = n. We have u0 u1 = zk−1 = v0 v1 . So, we have either (a) u0 = v0 and u1 = v1 , or, for some w, either (b) u0 w = v0 and u1 = w v1 , or (c) u0 = v0 w and w u1 = v1 . In case (b), we have: 0 ∗ w = 0 and

1 = w ∗ 1 . We leave (a) and (c) to the reader. Suppose k < n. We have:

0 ∗ (u1 zk · · · zn−2 v0 ) = 0 and 1 = (u1 zk · · · zn−2 v0 ) ∗ 1 . The case that k > n is similar. It is easy to see that ⊕ is a sum or coproduct in the sense of category theory. The following theorem is immediate. Theorem 8. If a is an atom of Ai , then a is an atom of A0 ⊕ A1 . Finally, we have the following theorem. Theorem 9. Let A be any set and let B := B, ∗ be any concatenation structure. We assume that A and B are disjoint. Then, there is an extension of B with at least A as atoms. Proof. Let A∗ be the free semi-group generated by A. We can take as the desired extension of B, the structure A∗ ⊕ B. Remark 5.1. The whole development extends with only minor adaptations, when we replace axiom TC2 by: ˙ (∃!w (x ∗ w = u ∧ y = w ∗ v) ∨ • x ∗ y = u ∗ v → ((x = u ∧ y = v) ∨ ∃!w (x = u ∗ w ∧ y ∗ w = v)) ˙ is exclusive or. Here ∨ §6. A canonical construction. Although we know that not every concatenation structure can be represented by decorated linear orderings, i.e., as a concrete concatenation structure, there may exist a canonical construction of a concrete concatenation structure which is a representation whenever there exists any concrete representation. In this section we shall propose such a construction, but we can only show that it is universal in a restricted subclass of all concrete representations. It will be now more convenient to work with a theory for monoids, rather than for semigroups, as we did in the previous sections. We will work in the theory TCε+ , which is TCε0 plus the following axiom. TCε 9. x ∗ y ∗ z = y → (x = ε ∧ z = ε). We do not postulate the existence of irreducible elements, as they do not play any role in what follows, but they surely can be present. We shall call elements of a model M of TCε+ : words. When possible, the concatenation symbol ∗ will be omitted. When considering representations of structures with a unit element ε by decorated order types, one has to allow an empty decorated order structure.

10

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

Thus a representation of a model M of TCε+ is a mapping that assigns a decorated order structure to every w ∈ M so that 1. (ε) = ∅, 2. (uv) = (u)(v) and 3. (w) ∼ = (v) implies w = v. Lemma 10. In a model M of TCε+ the binary relation ∃u(xu = y) deﬁnes an ordering on the elements of M. Deﬁnition 6.1. A k-partition of a word w is a k-tuple (w1 , . . . , wk ) such that w1 . . . wk = w; we shall often abbreviate it by w1 . . . wk . An ordering relation is deﬁned on 3-partitions of w by: • u1 u2 u3 ≤ v1 v2 v3 :⇔ ∃x1 , x3 (v1 x1 = u1 ∧ x3 v3 = u3 ). The axioms ensure that for any two partitions there is a unique common reﬁnement. Deﬁnition 6.2 (Word Ultraﬁlters). Let w be a word and S a set of 3partitions of w. We shall call S a word ultraﬁlter on w if 1. εwε ∈ S 2. xεy ∈ S for any x, y 3. if U ∈ S, V is a 3-partition of w and U ≤ V , then V ∈ S 4. if xyz ∈ S and y = y1 y2 , then exactly one of the following two cases holds: (x, y1 , y2 z) ∈ S or (xy1 , y2 , z) ∈ S. Let S be a word ultraﬁlter on w and xyz ∈ S. Then we deﬁne the natural restriction of S to y which is a word ultraﬁlter Sy on y deﬁned by: • (r, s, t) ∈ Sy :⇔ (xr, s, tz) ∈ S. We shall deﬁne an ordering on word ultraﬁlters on a ﬁxed w and an equivalence on word ultraﬁlters on all words of M . Let S and T be word ultraﬁlters on w, then we deﬁne: • S < T :⇔ ∃u, v ((ε, u, v) ∈ S ∧ (u, v, ε) ∈ T ). Let S and T be word ultraﬁlters on possibly diﬀerent words, then we deﬁne: • S ∼ T :⇔ ∃x, x , y, z, z (xyz ∈ S ∧ x yz ∈ T ∧ Sy = Ty ). Notice that < is a strict ordering on word ultraﬁlters on w, but for S < T it still may be S ∼ T . Deﬁnition 6.3. Let w be a word. The canonical decorated ordering associated with w is the ordering of all word ultraﬁlters on w, where each word ultraﬁlter S is decorated by [S]∼ , the equivalence class of ∼ containing S. This decorated ordering will be denoted by C (w). Here are some basic properties of C (w). • The topological space determined by the ordering is compact and totally disconnected. In particular, it has largest and smallest elements.

DECORATED LINEAR ORDER TYPES

11

• For every proper preﬁx x of w, there is a uniquely determined pair of word ultraﬁlters which forms a gap (no word ultraﬁlter in between). Thus there is a natural embedding of the ordering of the preﬁxes into C (w). More precisely, we have have two mappings φw− and φw+ such that for a proper preﬁx x the pair φw− (x), φw+ (x) is the gap corresponding to x. If x = ε (or x = w), then only φw+ (x) (or φw− (x)) is deﬁned and it is the least (largest) element of C (w). Furthermore the images of these mappings are dense sets in C (w). • Vice versa, every gap in C (w) corresponds to a preﬁx (or equivalently to a 2-partition). • If a is an atom (irreducible element in M), then it determines a principal word ultraﬁlter. For a given atom a all such principal word ultraﬁlters are equivalent. C satisﬁes conditions 1. and 2. but, in general, condition 3. fails. Example 6.4. Let M = A∗ be the monoid generated by A (the alphabet). Then, all word ultraﬁlters are principal and C (w) is essentially the string w itself. Example 6.5. This is a ‘pathological example’. Let M be the nonnegative real numbers with +. For a positive real r, the order type of C (r) is the order type of: {(0, 1)}∪(I ×{0, 1})∪{(1, 0)}, where I is the open unit interval, with the lexicographic order. The equivalence relation ∼ has two classes; elements of the form (x, 0) are decorated by one type of word ultraﬁlters, elements (x, 1) are decorated by the other type. Hence for every r, s > 0, C (r) ∼ = C (s), thus C is not a representation. Deﬁnition 6.6. is a regular representation of M by decorated orderings, if for every 2-partition x1 x2 = w of w ∈ M, there exists a unique 2-partition A1 A2 = (w) such that A1 ∼ = (x1 ) and A2 ∼ = (x2 ). We do not know if every concatenation structure that has a concrete representation also has a concrete regular representation. If is regular, we have an analogous property for k-partitions for every k. For a k-partition (x1 , . . . , xk ) of w in M, we shall write k (x1 , . . . , xk ) = (A1 , . . . , Ak ), where (A1 , . . . , Ak ) is the uniquely determined k-partition of (x1 . . . xk ) such that Ai ∼ = (xi ), for i = 1, . . . , k. Theorem 11. 1. If the canonical mapping C is a representation of M, then it is a regular representation of M. 2. If there exists a regular representation of M, then so is also C . Proof. Ad 1. Let uv = w be in M, and suppose we have two diﬀerent 2partitions AB = C (w), A B = C (w), with A = C (u) ∼ = A , B = C (v) ∼ = B .

12

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

Suppose that A is a proper initial segment of A . Since AB corresponds to a 2-partition uv, there is a gap between A and B. Since A is a proper initial segment of A , the gap is in A . As every gap corresponds to a 2-partition of the preimage in C , there exists y and D, such that A = AD and D ∼ = C (y). Hence u = uy, which is possible only if y = ε. But then D is empty, which is a contradiction. Ad 2. Given a regular representation we should show condition 3. for mapping C . Since is a representation it suﬃces to show that C (u) ∼ = C (v) implies (u) ∼ = (v). Our strategy is (i) to construct, for every w ∈ M , an order preserving mapping h with h : Supp((w)) → Supp(C (w)), and then (ii) to show that if : C (u) → C (v) is an isomorphism, then for every S ∈ Supp(C (u)) the ﬁbers of S and (S), as decorated orderings, are isomorphic, i.e., h −1 (S) ∼ = h −1 ( (S)), or they are both empty. If (i) and (ii) are true, then it is easy to construct an isomorphism (u) ∼ = (v): it suﬃces to connect the isomorphisms of the ﬁbers into one isomorphism. Ad (i). Let w ∈ M , let j ∈ Supp((w)). We deﬁne: • h(j) := {(x, y, z) | ∃A, B, D (j ∈ B and 3 (x, y, z) = (A, B, D))}. One can readily verify that h(j) is a word ultraﬁlter, and that h is order preserving. Ad (ii). Let S ∈ Supp(C (u)) and T ∈ Supp(C (v)) such that T = (S). Then S and T have the same decoration, which means that S ∼ T . By deﬁnition, there exist 3-partitions (x, y, z) ∈ S and (x , y, z ) ∈ T such that Sy = Ty . Let 3 (x, y, z) = (A, B, D) and 3 (x , y, z ) = (A , B , D ). Then, B∼ = B , as is a representation. Let us denote this isomorphism by κ. Take an arbitrary 3-partition y1 y2 y3 = y and let 5 (x, y1 , y2 , y3 , z) = (A, B1 , B2 , B3 , D), and 5 (x , y1 , y2 , y3 , z ) = (A , B1 , B2 , B3 , D ). Then Bi ∼ = Bi , for i = 1, 2, 3. By the regularity of , the segments B1 , B2 , B3 in B and the segments B1 , B2 , B3 in B are uniquely determined by their isomorphism types, whence: (1)

κ(Bi ) = Bi , for i = 1, 2, 3.

The ﬁber h −1 (S) is deﬁned as the intersection of all segments B2 that belong to 5-partitions (x, y1 , y2 , y3 , z) such that (y1 , y2 , y3 ) ∈ Sy = Ty . Similarly, the ﬁber h −1 (T ) is deﬁned as the intersection of all segments B2 that belong to 5-partitions (x , y1 , y2 , y3 , z ) such that (y1 , y2 , y3 ) ∈ Sy = Ty . According to (1), for all such 3-partitions, κ : (B1 , B2 , B3 ) ∼ = (B1 , B2 , B3 ). Hence κ is also −1 −1 an isomorphism of h (S) onto h (T ), or both ﬁbers are empty.

DECORATED LINEAR ORDER TYPES

13

REFERENCES

[1] John Burgess, Fixing Frege, Princeton Monographs in Philosophy, Princeton University Press, Princeton, NJ, 2005. [2] Andrzej Grzegorczyk, Undecidability without arithmetization, Studia Logica. An International Journal for Symbolic Logic, vol. 79 (2005), no. 2, pp. 163–230. [3] Laurence Kirby, Addition and multiplication of sets, MLQ. Mathematical Logic Quarterly, vol. 53 (2007), no. 1, pp. 52–65. [4] P. Komjath ´ and V. Totik, Problems and Theorems in Classical Set Theory, Problem Books in Mathematics, Springer, New York, 2006. [5] W. V. Quine, Concatenation as a basis for arithmetic, The Journal of Symbolic Logic, vol. 11 (1946), pp. 105–114. [6] W. Sierpinski, Cardinal and Ordinal Numbers, Polska Akademia Nauk, Monograﬁe ´ ´ Matematyczne, vol. 34, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958. [7] A. Tarski, Der wahrheitsbegriﬀ in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1935), pp. 261– 405, Reprinted as [9]. The paper is a translation of the Polish Poje¸cie prawdy w je¸zykach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego, Wydział III matematyczno-ﬁzycznych, no. 34, Warsaw 1933. [8] , The notion of rank in axiomatic set theory a some of its applications. abstract 628, Bulletin of the American Mathematical Society, vol. 61 (1955), p. 433. [9] , The concept of truth in formalised languages, Logic, Semantics, Metamathematics (A. Tarski, editor), Oxford University Press, Oxford, 1956, (this paper is a translation of [7]), pp. 152–278. [10] A. Visser, Growing commas—a study of sequentiality and concatenation, Notre Dame Journal of Formal Logic, vol. 50 (2009), no. 1, pp. 61–85. DEPARTMENT OF MATHEMATICS ˇ UNIVERSITY OF ZAGREB, BIJENICKA 30 10000 ZAGREB, CROATIA

E-mail: [email protected] MATHEMATICAL INSTITUTE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC ˇ ´ 25, 115 67 PRAHA 1, CZECH REPUBLIC ZITN A

E-mail: [email protected] SCHOOL OF PHILOSOPHY, ANTHROPOLOGY AND SOCIAL INQUIRY THE UNIVERSITY OF MELBOURNE PARKVILLE 3010, AUSTRALIA

E-mail: [email protected] DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TORONTO, TORONTO ONTARIO, CANADA M5S 1A4

E-mail: [email protected] DEPARTMENT OF PHILOSOPHY UTRECHT UNIVERSITY, HEIDELBERGLAAN 8 3584 CS UTRECHT, THE NETHERLANDS

E-mail: [email protected]

CARDINAL PRESERVING ELEMENTARY EMBEDDINGS

´ EDUARDO CAICEDO ANDRES

Abstract. Say that an elementary embedding j : N → M is cardinal preserving if CARM = CARN = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V . We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M .

§1. Introduction. This paper is the ﬁrst of a series attempting to investigate the structure of (not necessarily ﬁne structural) inner models of the set theoretic universe under assumptions of two kinds: 1. Forcing axioms, holding either in the universe V of all sets or in both V and the inner model under study, and 2. Agreement between (some of) the cardinals of V and the cardinals of the inner model. I try to be as self-contained as is reasonably possible, given the technical nature of the problems under consideration. The notation is standard, as in Jech [8]. I assume familiarity with inner model theory; for ﬁne structural background and notation, the reader is urged to consult Steel [19] and Mitchell [15]. In the remainder of this introduction, I include some general observations on large cardinal theory, forcing axioms, and ﬁne structure, and state the main results of the paper. ¨ Consider set theory with the axiom of choice as formalized by the GodelBernays axioms GBC, so we can freely treat proper classes. An inner model (or simply, a model) is a transitive class model M of the Zermelo-Fraenkel ZFC axioms containing all the ordinals. If M is a model and ϕ is a statement, ϕ M is the assertion that ϕ holds in M . If is a deﬁnable term, M indicates the interpretation of inside M . Denote by ORD the class of ordinals and by CAR the class of cardinals. The coﬁnality of an ordinal α is denoted cf(α). All our embeddings are elementary and non-trivial, and the classes involved are inner models; the critical point of such an embedding j is denoted cp(j). Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

14

CARDINAL PRESERVING ELEMENTARY EMBEDDINGS

15

1.1. Large cardinal axioms. See Kanamori [10] as a general reference for large cardinals. It is well-known that the GBC axioms fall quite short of providing a complete picture of the universe of sets. Among the many independent extensions that have been studied, two sets of statements have been isolated as natural candidates to add to the basic axioms: large cardinal axioms and forcing axioms. By far, large cardinal axioms are better understood and more readily accepted. There is no formal deﬁnition of what a large cardinal is, but a few features can be distinguished. Typically, they are regular cardinals κ such that Vκ is itself a model of ZFC and, more importantly for present purposes, one can associate to κ a family of elementary embeddings j : V → M where M is a transitive class. The association is usually (as in the case of measurable, strong or supercompact cardinals) that κ is the critical point cp(j) of j, the ﬁrst ordinal α such that j(α) > α, but it can take other shapes, as is the case with Woodin cardinals. An important remark is that these notions can be stated in terms of the existence of certain ultraﬁlters or systems of ultraﬁlters called extenders, the connection being given by an analogue of the model theoretic ultrapower construction. An extender is essentially (a family of ultraﬁlters coding together) a fragment of an elementary embedding, and it is by now a standard device; good expositions and deﬁnitions can be found in Jech [8, Chapter 20], Kanamori [10, § 26], and Steel [19, § 2.1], among others. Brieﬂy: For a set X and a cardinal κ, let [X ]κ = {Y ⊆ X : |Y | = κ} and deﬁne [X ] κ. A non-trivial (κ, )extender E is a sequence (Ea : a ∈ [] n(. . . )’, so the computational challenge of the theorem is to ﬁnd a realizer or bound for ∃n. If we furthermore restrict the ε > 0 to those that can be written as 2−k for k ∈ N, we see that the theorem is almost purely arithmetical — except for parameters f and T which involve the types X . In a moment, we shall see how to cast even those parameters in a such a way that they only contribute to the computational challenge of the theorem by natural numbers. This theorem can be formalized in the formal system A [X, ·, ·]. The theory A is Peano arithmetic extended to all ﬁnite types and extended with

PROOF MINING IN PRACTICE

85

the axiom schema of dependent choice. Most of classical analysis can be formalized in this theory. The theory A [X, ·, ·] denotes the extension of A with an abstract Hilbert space in the following way: • We add a new ground type X for the elements of the Hilbert space and extend A to all ﬁnite types T X over N and X . • We add new constants representing the inner product and the vector space operations on X . • We add new axioms describing the algebraic properties of the inner product and the vector space operations. ¨ Using a monotone variant of Godel’s Dialectica interpretation, one proves general metatheorems about the extraction of bounds from proofs of ∀∃statements in A [X, ·, ·] and similar theories. For details, see [2]. The Mean Ergodic Theorem is a ∀∃∀-statement. As mentioned above, such theorems do not always allow to extract realizers or bounds for the existential quantiﬁer. This is also the case with the Mean Ergodic Theorem, where we can construct a Hilbert space and a mapping T such that a full rate of convergence for the ergodic averages An f would solve the halting problem for Turing machines. Even in the measure theoretic setting, one may construct counterexamples. Only when the measure space is ergodic, one obtains a full rate of convergence. For details, see [1]. Instead, we will extract bounds for the classically equivalent but constructively weaker no-counterexample version of the Mean Ergodic Theorem: Mean Ergodic Theorem (no-counterexample version). Let (X, ·, ·) be a Hilbert space, let T : X → X be a linear, nonexpansive selfmapping of n 1 i X . For f ∈ X deﬁne An f := n+1 i=0 T f, then for every ε > 0 and and every number-theoretic function M : N → N there exists an n ∈ N s.t. AM (n) f − An f < ε. The function M claims the existence of a counterexample to the convergence of An f, and the existence of n shows that any such supposed counterexample can be refuted. This no-counterexample version is classically equivalent to full convergence. Constructively, this establishes the local stability (up to an ε > 0) of the ergodic averages An f for arbitrarily long periods of time. The theorem is now in a suitable form for the metatheorems mentioned above to be applicable. From the most general form of the metatheorems in [2] we derive the following special case: Deﬁnition 2. The ﬁnite types T X are deﬁned as follows: (i) N, X ∈ T X , (ii) , ∈ T X ⇒ → ∈ T X . We often write 0 for the type N and 1 for the type N → N. Deﬁnition 3. A formula F is called a ∀-formula, resp. ∃-formula, if it is of the form ∀a Fqf , resp. ∃a Fqf , where Fqf is quantiﬁer-free and the types of

86

PHILIPP GERHARDY

are small1 , which includes amongst others the types N, X , N → N, N → X and X → X . We write F∀ and F∃ for ∀-formulas and ∃-formulas respectively. Corollary 4. Let (X, ·, ·) be an abstract Hilbert space. If ∀f X , T X →X , k 0 , M 1 T n.e. ∧ ∀u 0 B∀ (f, T, k, M, u) → ∃v 0 C∃ (f, T, k, M, v) is provable in A [X, ·, ·], then there is a computable φ : N × N × NN → N such that ∀f X , T X →X , k 0 , M 1 f ≤ b ∧ T n.e.∧ ∀u 0 ≤ φ(b, k, M )B∀ → ∃v 0 ≤ φ(b, k, M )C∃ holds in any Hilbert space (X, ·, ·). This is exactly the logical form of the no-counterexample version of the Mean Ergodic Theorem, and it is easy to see that the standard proof of the (no-counterexample version) Mean Ergodic Theorem can be formalized in A [X, ·, ·], as the proof only uses the basic algebraic properties of normed spaces and inner products. Thus, this corollary predicts a computable bound on ∃n in the Mean Ergodic Theorem in terms of (an integer bound b on) f, ε and the counterexample function M , but independent of the particular f (as long as its norm is bounded by b), the particular T or the particular space (X, ·, ·). In [1], the following bounds were obtained:

ik 2lb b nk = M i0 = 0, ε2 ε l =0 15 4 4 2 M (nk ) b ik + 1 = ik + ε4 2

Let d = 512b , then for some n ≤ N (b, ε, M ) = 2nεd b , we have that AM (n) f − ε2 An f < ε. In the next sections, we discuss details of the proof analysis carried out in [1] which resulted in these bounds. §3. Interpreting the law of the excluded middle. The main argument of the proof of the Mean Ergodic Theorem goes as follows: Any element f ∈ X can be written as a sum f = h + g, where h is T -invariant (i.e. Th = h) and g can be approximated arbitrarily well by an element of the form u − Tu with u ∈ X . Since T is linear, we may thus study An h and An g seperately. Obviously, An h = h for all n ∈ N. Also, An (u − Tu) ≤ u 2n , which quickly converges to 0. In other words, the main argument is: Either f −Tf = 0, f is T -invariant and An f is constant, or f −Tf > 0 and then An f converges, because the part that is not T -invariant tends to 0 quickly. To estimate how 1 For

the exact deﬁnition, see [2].

PROOF MINING IN PRACTICE

87

fast An (u − Tu) converges to 0, we need to know the norm u, or at least an upper bound on u. As the decomposition of f into g and h corresponds to a decompostion of the whole space X into orthogonal subspaces U (the closure of the space of elements of the form u − Tu) and V (the subspace of T -invariant elements), this is equivalent to approximating the projection of f onto U by some u − Tu and obtaining an upper bound on u. As the averages An f lie in the subspace generated by {f, Tf, T 2 f, . . . }, it suﬃces to consider the projection of f onto Uf = {T i f − T i+1 f|i = 0, 1, 2, . . . }. For this subspace Uf ⊆ U , we can explicitly deﬁne a sequence un such that gn = un − Tun converges to the projection of f onto Uf , and it is from this sequence that we derive an upper bound on the norms un . The elements un are deﬁned as follows: f, f − Tf f, f − Tf2 f − (ui − Tui ), T i+1 f − T i+2 f i+1 = T f. T i+1 f − T i+2 f2

u0 = ui+1

The numerator of the fraction has an easy upper bound by f, f − Tf ≤ ff−Tf, resp. f−(ui −Tui ), T i+1 f−T i+2 f ≤ fT i+1 f−T i+2 f. Thus we only need to ﬁnd a lower bound for f − Tf, resp. T i+1 f − T i+2 f in the denominator — but the norm of elements T i f − T i+1 f can be arbitrarily small. The solution lies in the original argument: If the norm of T i f − T i+1 f is zero for some i, then T i f is T -invariant and we are done. If not, it must be larger than some > 0. This is close to the solution, but not quite there yet. As we are only looking for local stability for An f, not full convergence, it suﬃces that T i f − T i+1 f is small enough, as then repeated use of the triangle inequality yields local stability. Given ε > 0 and the counterexample function M , we can say how small T i f − T i+1 f needs to be to directly imply local stability, i.e. we can produce an explicit > 0 such that T i f − T i+1 f < yields the result. Otherwise, we have a lower bound on T i f − T i+1 f and therefore an upper bound for ui ! In [1], we prove the following lemma (Lemma 2.15 in [1]), reﬂecting the above discussion: Lemma 5. For any i ≥ 0 and ε > 0, either 1. there is an n ≤ 2i f ε such that AM (n) f − An f ≤ ε, or f2 i f 2. ui ≤ 2ε j=0 M (2j ε ). In summary, we use the fact that for any i ∈ N either T i f − T i+1 f is zero or it is not zero. We give a computational interpretation of this instance of the law of the excluded middle by producing, for each i ∈ N, a > 0 such that both T i f − T i+1 f ≤ and T i f − T i+1 f > yield an n ∈ N such

88

PHILIPP GERHARDY

that AM (n) f − An f < ε. The maximum of the two is then an upper bound on an n ∈ N such that the result holds. §4. The principle of convergence for monotone sequences. Another important aspect of the proof is the use of the convergence of the sequence gn = un − Tun to the projection of f onto Uf . The convergence of gn is established in the following way: For every n ∈ N, the norm gn ≤ f. Also, gn ≤ gn+1 , so the sequence an = gn is a bounded monotone sequence of real numbers, and hence it is a Cauchy sequence. A simple argument shows that for every > 0 there is a > 0 such that if |ai − ai+d | < then gi − gi+d < , so that we can obtain a rate of convergence for gn from a rate of convergence for an . The convergence of gn is used to give an estimate of AM (n) f − An f. For any i, n ∈ N: AM (n) f − An f ≤ AM (n) (f − gi ) − An (f − gi ) + AM (n) gi + An gi . One then shows that there is > 0 such that if gi − gi+d < for all d > 0 then AM (n) (f − gi ) − An (f − gi ) < ε/2 for all n. With an upper bound on ui (such as the one we sketched in the previous section), we can ﬁnd an n large enough so that AM (n) gi , An gi < ε/4. Combined, this yields AM (n) f − An f < ε. Classically, the full convergence of the sequence gn is established using the principle of convergence for monotone sequences, but constructively, we cannot obtain a full rate of convergence for the gn , as the convergence of gn and the ∀∃∀ version of the Mean Ergodic Theorem are equivalent constructively. However, we may still give a computational interpretation of this particular appeal to the principle of convergence for monotone bounded sequences in the proof of the Mean Ergodic Theorem. As this may sound confusing, here is another explanation: We cannot obtain a computable rate of convergence, but we can give a computational interpretation to the use of its existence as a lemma. This is because a proof of a ∀∃-theorem cannot fully exploit the computational strength of a ∀∃∀-lemma. The principle of convergence for monotone bounded sequences is an instance of arithmetical comprehension. The computational interpretation of arithmetical comprehension in general requires bar recursion (see [6]). Even though comprehension is applied to a sequence an = gn which is explicitly given in the parameters of the proof, a computational interpretation of this instance of comprehension — although simpler — would still vastly increase the complexity of the extracted bounds. Instead we opt for a diﬀerent approach. In [3], Kohlenbach describes a technique to eliminate certain instances of arithmetical comprehension without increasing the growth rate (relative to the Grzegorczyk classes of computational complexity) of the bounds extracted by monotone functional interpretation. In this particular case, the informal

89

PROOF MINING IN PRACTICE

idea amounts to this: Although we obtain the Mean Ergodic Theorem from the full convergence of gn , we only need to establish gi − gi+d < for a particular d which is expressed in the other parameters and a hypothetical i. The exact d can be read oﬀ of the proof — as made explicit in the lemmas stated below — and from this we may form a sequence of non-overlapping intervals [ik , ik +dk ] where i0 = 0 and ik+1 = ik +dk . Now recall that there was a > 0 such that |ai − ai+d | < implies gi − gi+d < . Since the sequence an is monotone and bounded |aik − ajk | cannot exceed > 0 inﬁnitely often. Hence, |aik − ajk | < for some k ∈ N, and the result follows. In total, this amounts to replacing the use of full convergence with the statement that there is no counterexample to convergence, and the latter has, as sketched, a simple primitive recursive interpretation. Interestingly, the usefulness of this computationally weaker, no-counterexample version of convergence was recently (re-)discovered independently by Terrence Tao [7] under the name “Finite convergence principle”. Tao uses this principle to establish ﬁnitary, combinatorial proofs of some convergence results in ergodic theory, but without proving explicit bounds. See [8] for details. In [1], this argument yields the following sequence of lemmas, concerned with establishing AM (n) (f − gi ) − An (f − gi ) ≤ ε/2 by making gi − gi+d small enough for some i and d : 4

. Then for every i there is a j Lemma 6. Let ε > 0, let d = d (ε) = 32f ε4 in the interval [i, i + d ) such that T (f − gj ) − (f − gj ) ≤ ε. 4

4

Lemma 7. Let ε > 0, let n ≥ 1, let d = d (n, ε) = d (2ε/n) = 2n εf . 4 Then for every i there is a j in the interval [i, i + d ) such that An (f − gj ) − (f − gj ) ≤ ε. Lemma 8. Let ε > 0, let m ≥ 1, let d = d (m, ε) = d (m, ε/2) = 4 4 32mεf . Furthermore suppose gi − gi+d ≤ ε4 . Then for any n ≤ m, 4 An (f − gi ) − (f − gi ) ≤ ε. Lemma 9. Let ε > 0, let m ≥ 1, let d = d (m, ε) = d (m, ε/2) = 4 4 2 m εf . Furthermore suppose gi − gi+d ≤ ε8 . Then for any n ≤ m, 4 An (f − gi ) − Am (f − gi ) ≤ ε. 9

The last three lemmas follow fairly easy from the ﬁrst one, repeatedly appealing to the triangle inequality. The ﬁrst is proved using general properties of the inner product in Hilbert spaces; see [1] for details. The last lemma shows exactly, how the convergence of gn allows us to bound An (f−gi )−Am (f−gi ), where eventually m is replaced by M (n). In other words, this insight allows us to prove the above lemmas and later the no-counterexample convergence of An f already from the no-counterexample convergence of gn : The expression deﬁning dk in terms of ik and the other parameters is nothing but a counterexample function. In this case, there is

90

PHILIPP GERHARDY

a nested appeal to the no-counterexample convergence of gn , resp. an , as dk needs to be big enough enough to ensure that |adk − adk +1 | < k where k is given in terms of ik and the other parameters. Thus sk depends on ik , while ik depends on previous values dk−1 and thereby on ik−1 . The required > 0 such that gik − gik +dk < implies AM (n) f − An f ≤ ε, and thus the number of intervals [ik , ik + dk ] we need to consider, is independent of k though. Thus we consider longer and longer intervals [ik , ik + dk ], but only a ﬁxed number of those. For the details, see [1]. §5. Conclusions. In the previous three sections, we illustrated three aspects of a recent application of proof mining to the Mean Ergodic Theorem. At the heart of this proof analysis are proof theoretic ideas that originate from the general logical metatheorems in [2]. On the surface, this analysis is based on ideas not dependent of particular aspects of mathematical logic: (1) making the computational meaning or challenge of a theorem explicit, (2) giving a computational interpretation of an instance of the law of the excluded middle, and (3) reﬁning the appeal, within the proof, to the principle of convergence for monotone bounded sequences of real numbers. In this way, the author hopes that the above examples illustrate that proof mining yields analyses and reﬁnements of mathematical proofs that ought to be considered mathematical even by those who do not consider logic and proof theory proper mathematics. The proof theoretic perspective on mining proofs merely provides a systematic tool to carry out these analyses and to guide intuition where the result may not be obvious otherwise. Acknowledgements. The author would like to thank the reviewers for making helpful suggestions for improving the presentation in this paper.

REFERENCES

[1] J. Avigad, P. Gerhardy, and H. Towsner, Local stability of ergodic averages, to appear in Transactions of the American Mathematical Society. [2] P. Gerhardy and U. Kohlenbach, General logical metatheorems for functional analysis, Transactions of the American Mathematical Society, vol. 360 (2008), no. 5, pp. 2615–2660. [3] U. Kohlenbach, Elimination of Skolem functions for monotone formulas in analysis, Archive for Mathematical Logic, vol. 37 (1998), no. 5-6, pp. 363–390, Logic Colloquium ’95 (Haifa). [4] , Some logical metatheorems with applications in functional analysis, Transactions of the American Mathematical Society, vol. 357 (2005), no. 1, pp. 89–128. [5] , Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008. [6] C. Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proceedings of symposia in pure mathematics, AMS, Providence, 1962, pp. 1–27. [7] T. Tao, Soft analysis, hard analysis, and the ﬁnite convergence principle, On T. Tao’s blog: terrytao.wordpress.com.

PROOF MINING IN PRACTICE

91

[8] , Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynamical Systems, vol. 28 (2008), no. 2, pp. 657–688. [9] A. S. Troelstra (editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973. DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO BLINDERN, N-0316 OSLO, NORWAY

E-mail: [email protected]

CARDINAL STRUCTURE UNDER AD

STEVE JACKSON

Our aim in this paper is to survey the theory of cardinal structure assuming AD, the axiom of determinacy (deﬁned below). We work throughout in the theory ZF + DC + AD. The axiom of determinacy was introduced by Mycielski and Steinhaus in the early 60’s and quickly developed into a powerful tool for extending the ZF results of the classical descriptive set theorists about Σ11 (analytic) and Π11 (co-analytic) sets to higher levels of the projective hierarchy. The axiom contradicts the axiom of choice AC, but it was understood that it should be applied in a restricted universe such as L(R) (the smallest model of set theory containing the reals) where it seemed like a reasonable axiom. It wasn’t until much later, through the work of Martin, Steel, and Woodin (see [12], [20]), that it was shown that ZFC plus large cardinal assumptions actually imply that AD holds in L(R). The work of the Cabal from the late 60’s through the 80’s developed an extensive theory of pointclasses and associated properties from this axiom. A good reference for those developments is Moschovakis’ book [13] and the Cabal volumes themselves ([8], [9], [18], and [4]). This theory of the projective sets (and beyond) was largely developed in terms of certain ordinals called the projective ordinals, the 1n (Deﬁnition 1.12 below). It was also realized that AD had much to say about the properties of cardinals in general. An example would be Solovay’s early result that ℵn is singular with coﬁnality ℵ2 for n ≥ 3 (see Theorem 8.2 of [8]). The theory at the time, however, was not suﬃcient to calculate the values of the 1n for n ≥ 5 nor to develop the theory of the cardinals past ℵ . Kunen and Martin independently discovered the idea of a homogeneous tree. The precise concept and deﬁnition was formulated by Kechris. Using this concept, Kunen initiated a program for computing the 1n . The program stalled, however. Kechris’ article [8] gives an exposition of what was known at this time. In the early 80’s Martin proved a result which in current terminology showed the existence of the Martin tree. This generalized the earlier construction of the Kunen tree which played an important role in the AD theory of the ℵn . Building on this and some joint work with Martin, the author computed the 1n and developed 2000 Mathematics Subject Classiﬁcation. 03E60, 03E55, 03E05. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

92

CARDINAL STRUCTURE UNDER AD

93

a theory for analyzing the cardinal structure through the supremum of the 1n and a ways beyond. In particular, this theory of the projective ordinals generalizes in a straightforward manner to determine the cardinal structure through ℵ1 (the supremum of the 1n is ℵ0 , where 0 = supn (n) where we set (0) = 1 and (n + 1) = (n) ). In hindsight, a key concept that was missing in the earlier theory was the notion of a description. These are ﬁnitary objects which, roughly speaking, describe how to generate ordinals through the iterated ultrapowers of certain canonical measures. The earlier theory can in fact be viewed as a very simple case of the general theory, using only “trivial descriptions” (which are just integers). This point is explained in more detail in [3]. It turn out that the descriptions in fact completely describe the cardinal structure. One of the main goals of this paper is to introduce the descriptions in as simple a manner as possible, and to show how they determine the cardinal structure. In the complete analysis one must, in addition to deﬁning the descriptions, analyze the measures (countably additive ultraﬁlters) on the cardinals and establish certain partition properties (the strong partition property on the 12n+1 ) as well as verify certain inductive hypotheses. These additional arguments, however, use the same notion of description. In this paper, we focus on the descriptions themselves and the cardinal structure they generate. The reader wishing to see the details of the complete inductive analysis can consult [5] for the complete ﬁrst step of the inductive analysis (including the computation of 15 ) and [4] for the general step. The reader can also consult [3] for a more complete exposition of how descriptions are used in other ways (e.g., in analyzing the measures and establishing the partition properties). Knowledge of these other papers is not necessary for this paper, however. Rather than trying to explain in detail how descriptions are used in the complete inductive analysis, we introduce them here through two “exemplary problems.” These two problems can be stated entirely down at the level of the ℵn , but require the development of the same descriptions needed to compute 15 . As is turns out, 15 = ℵ +1 , and the descriptions we introduce through these two problems suﬃce to analyze the cardinal structure below this point. Through this approach, the reader can see clearly the underlying combinatorics in a much simpler setting. Our aim is that reader with a knowledge of basic determinacy theory at the level of [13] and perhaps [8] can follow the main thread of this paper. The reader could consult [10] or [13] for more general background on descriptive set theory. In §1 we give some background and sketch some of the “global” theory of AD. By this we mean the more general results that hold for all pointclasses and are independent of the more detailed analysis using the descriptions. This section is deﬁnitely of a survey nature and is included for the sake of completeness. In §2 we give a brief sketch of the overall plan of how the

94

STEVE JACKSON

inductive analysis of the projective ordinals goes. This section is also included for the sake of completeness and to give the reader a sense of how descriptions ﬁt into the bigger picture. The later sections do not speciﬁcally depend on these sections, and the reader wishing to quickly see the notion of description can skip them. In §3 we recall some basic deﬁnitions and facts about partition relations. Partition relations play a central role in how descriptions are used to generate ordinals. Our approach in this paper is to assume the strong partition property on the odd projective ordinals, the 12n+1 , and proceed straight to the descriptions. In the actual full analysis, the descriptions are also used to prove the partition relations. In §4 we introduce the two canonical families of measure W1m and S1m and the ﬁrst exemplary problem: compute the ultrapower jS1m (n ) of the cardinal n by the measure S1m . Solving this problem will only need the introduction of the “trivial” descriptions (actually slight generalizations of these). In §5 we consider the second exemplary problem: compute the iterated ultrapower jS m1 ◦ jS m2 ◦ · · · ◦ jS1mt (n ). The 1 1 answer is actually immediate from the solution to the ﬁrst problem, however we can describe the cardinal structure in the iterated ultrapower by introducing the next level of description. These are the same objects one uses in the full analysis to compute 15 , prove the strong partition relation on 13 , and the weak partition relation on 15 (which is the ﬁrst step of the inductive analysis). We carry out an example in computational detail. In §6 we sketch how these descriptions actually are used to generate the cardinal structure between 13 = ℵ+1 and 15 = ℵ . We also give a brief hint at what goes into the more general deﬁnition of description. In §7 we give an entirely diﬀerent mechanism which can be used to present the cardinal structure below the projective ordinals. This is done via a certain algebra which is deﬁned directly and does not need descriptions. Descriptions are needed, however, to prove that this alternate formulation works (these proofs are not given here). This ¨ [7] and extends earlier notational framework involves joint work with B. Lowe joint work of the author and F. Khaﬁzov [6]. Thus, although the proofs that this method works are quite involved, this gives a direct and self-contained mechanism for understanding the cardinal structure. This should make the techniques of this area accessible to a wider audience. Finally, in §8 we present as an application of the theory of cardinal structure under AD a result concerning the collapse of cardinals from L(R) to V . Namely, we sketch a proof of the fact that assuming large cardinals in V plus the saturation of the non-stationary ideal on 1 , some regular cardinal in L(R) below (ℵ2 )L(R) is collapsed in V . The proof we present here is only a sketch, as complete details will be given elsewhere. This section is not intended to be self-contained, and is included to give an example of how the theory of the cardinal structure under AD can be used to get results in the ZFC world.

95

CARDINAL STRUCTURE UNDER AD

§1. Basic concepts. Though our intention is to eventually specialize to results of a “local” nature, that is, pertaining to the smaller cardinals for which the inductive combinatorial methods work, we present as background here AD results of a more general nature. There results are “global” in nature in that they hold throughout the Wadge hierarchy under AD. The reader already comfortable with the basics of AD theory can skip this section, or skim it to see our (mostly standard) notation. Let denote the Baire space with the usual topology, i.e., the product of the discrete topology on . is homeomorphic to the set of irrationals (as a subspace of R), and as is customary, we often call the “reals.” We let Σ01 denote the collection of open sets, and Π01 the collection of closed sets in . The levels of the Borel hierarchy are deﬁned as usual by A ∈ Σ0α (for α < 1 ) iﬀ A = n∈ An where An ∈ Π0αn for some αn < α. Also, A ∈ Π0α iﬀ Ac ∈ Σ0α (we use Ac to denote − A). The collection of Borel sets is 0 1 α 0, and take M0 to be the real algebraic numbers. The family {(0, α ) : α ≡M0 α} has the ﬁnite intersection property but (0, α) ∩ M0 is empty. 3. In the stable case, the analogous theorem to Theorem 5.7 is true for any deﬁnable set because the assumption is equivalent to the forking of X (a) over M0 . 4. The description of a deﬁnably compact set using a type-deﬁnable open covering (see Fact 2.3) easily follows from Fact 5.7. End of Digression. Proof of Theorem 5.3. Because G is aﬃnely embedded it is closed and bounded in M k . Assume X ⊆ G is not left generic. By Fact 5.4 we may assume that X is closed. Fix M0 such that X is deﬁnable over M0 . By Fact 5.6, there exists g ∈ G such that M0 ∩ Xg = ∅. By Theorem 5.7, there are g1 , . . . , gk (each realizing the same type as g over M0 ) such that k i=1

Xgi = ∅.

PILLAY’S CONJECTURE

191

By taking complement we get k

(G \ X )gi = G.

i=1

Hence, G \ X is right-generic.

Remarks. 1. Theorem 5.3 fails without the deﬁnable compactness assumption: The set (a, +∞) and its complement are both not generic in R, + (here left and right-generic are the same). 2. The analogue of Theorem 5.3 in the stable setting is true for any deﬁnable subset of the group G. 3. Recently, Eleftheriou has pointed out that the assumption that G is aﬃnely embedded can be omitted Theorem 5.7 by working in the manifold charts of G. Here are two easy consequences: Fact 5.8. Assume that G is deﬁnably compact and abelian. (1) The non-generic sets form an ideal. (2) Every formula deﬁning a generic set in G belongs to a complete “generic” type p (over M). Namely, every formula in p deﬁnes a generic set in G. §6. Some theory of generic sets II: Measure and the NIP. The content of Sections 6-9 is mostly taken from [18]. The connection between the Non Independence Property and measure is due to Keisler [21] and the proof of Lemma 6.4 below is modeled after a proof from Keisler’s paper. The next notion is due to Shelah. The deﬁnition we use is equivalent to the original one. Deﬁnition 6.1. A theory T is said to be dependent or to have the Non Independence Property (NIP) if for every indiscernible sequence ai : i < over A and φ(x, y) a formula over A the type {φ(x, b2j ) φ(x, b2j+1 ) : j < } is inconsistent. (We take φ to mean (φ ∧ ¬) ∨ (¬φ ∧ )) Stable theories, o-minimal theories, the theory of p-adically closed ﬁelds all have the NIP, while the theory of pseudo-ﬁnite ﬁelds fails to have it. In fact, by Shelah’s work any simple unstable theory fails to have NIP. Deﬁnition 6.2. We say that G admits a left invariant Keisler measure if there exists a real valued ﬁnitely additive measure : Def(G) → R on the deﬁnable subsets of G, such that (G) = 1 and for every deﬁnable X ⊆ G and g ∈ G, we have (gX ) = (X ). In the rest of this section we make the following assumptions on the group G (equipped with the deﬁnable sets induced by the ambient structure):

192

YA’ACOV PETERZIL

• G has NIP. • The non left-generic sets form an ideal. • G admits a left-invariant Keisler measure. As we will eventually show, every deﬁnably compact group in an o-minimal structure satisﬁes all of the above. For now, notice that any abelian deﬁnably compact group satisﬁes the above assumptions. Indeed, o-minimality implies NIP, and by Fact 5.8 the non-generic sets form an ideal. Because every abelian group is amenable, it admits a left-invariant real valued ﬁnitely additive probability measure on all subsets. Deﬁnition 6.3. For X, Y ⊆ G deﬁnable, we write X ≈ng Y if X Y is not left-generic. Notice that since we assume that the non left-generics form an ideal ≈ng is an equivalence relation. The NIP assumption is crucial for the following. Lemma 6.4. The equivalence relation ≈ng is bounded. I.e., there exists a ﬁxed small model M0 such that every equivalence class of ≈ng is already represented by a deﬁnable set over M0 . Proof. Let denote the ﬁnitely additive left-invariant measure on Def(G), the family of deﬁnable subsets of G. Note that if X ⊆ G is a deﬁnable ngeneric set then we have (X ) 1/n. Assume that ≈ng is unbounded. Then there exists a formula φ(x, y) over the empty set, with the variable x for elements in G, and unboundedly many bi ’s, such that φ(G, bi ) φ(G, bj ) is generic. By standard Ramsey-type arguments, there exists a ﬁxed n and an indiscernible sequence ai : i < such that for every i = j, the set φ(G, ai ) φ(G, aj ) is n-generic. Consider the family F = {Yj = φ(G, a2j ) φ(G, a2j+1 ) : j < }. By indiscernibility, there exists a natural number k such that every k sets in F have empty intersection. However, for every j, (Yj ) 1/n, and because (G) = 1 it is impossible that every k sets in F intersect trivially. Contradiction. Deﬁnition 6.5. For X ⊆ G deﬁnable, let Stabng (X ) = {g ∈ G : gX ≈ng X }. Under our assumptions, the set Stabng (X ) is a subgroup of G. It is typedeﬁnable because for every n, the set of all g such that n translates of gX X do not cover G, is deﬁnable. The map g → gX/ ≈ng induces an injective map from G/Stabng (X ) into Def(G)/ ≈ng and therefore we proved: Theorem 6.6. For any deﬁnable X ⊆ G, the subgroup Stabng (X ) has bounded index in G. §7. The proof of PC in the abelian case. We assume in this section that M expands a real closed ﬁeld and that G is deﬁnably compact and abelian.

PILLAY’S CONJECTURE

193

Our goal here is to prove: Theorem 7.1. If G is deﬁnably compact and abelian then dimLie (G/G 00 ) = dimM G. Proof. Because G 0 has ﬁnite index in G we may assume that G is deﬁnably connected. The proof is based on two ingredients. The ﬁrst one is a deep theorem of Edmundo and Otero on the torsion points in deﬁnably connected, deﬁnably compact abelian groups. (Presumably, this was one of the main justiﬁcations to the original conjecture of Pillay). Its proof is based on Cohomological tools and uses extensively the triangulation theorem which is true only in o-minimal expansions of real closed ﬁelds: Theorem 7.2. [13] Assume that M expands a real closed ﬁeld and that G is a deﬁnably compact, deﬁnably connected abelian group of dimension n. Then Tor(G) Tor(Tn ). (where Tn is the n-dimensional torus and Tor(G) is the subgroup of torsion elements in G). The second ingredient, which we will prove below is: Lemma 7.3. G 00 is torsion-free. Let us see how the two results, taken together, imply PC in the abelian case: Lemma 7.3, together with the divisibility of G 00 (see Corollary 4.2 (1)) imply that Tor(G/G 00 ) Tor(G). If dim(G) = n, it follows from theorem [13], that Tor(G/G 00 ) Tor(Tn ). Because G/G 00 is a connected (Theorem 4.1) abelian compact Lie group, it is Lie isomorphic to a direct sum of T1 ’s. The number of these T1 ’s is determined by, say, the number of 2-torsion points, therefore G/G 00 (T1 )n and so the real dimension of G/G 00 is n. Proof of Lemma 7.3. For every n ∈ N, consider the map n : g → g n from G onto G. By deﬁnable choice, there exists a deﬁnable set X ⊆ G such that n |X is a bijection of X and G (we assume that G is deﬁnably connected). By [13] (or actually by the preceding results in [37]), Tn = ker( n ) is ﬁnite. It clearly contains all n-torsion points and, as easily checked, G equals a ﬁnite disjoint union of the gX ’s, for g ∈ Tn . Thus X and all the gX ’s are generic and pair-wise disjoint, and therefore Tn ∩ Stabng (X ) = {0}. Because this is true for every n, the group Stabng (X ) is torsion-free. By Theorem 6.6, Stabng (X ) has bounded index in G, and therefore G 00 ⊆ Stabng (X ). It follows that G 00 is torsion-free, ending the proof of Lemma 7.3 and thus PC in the abelian case.

194

YA’ACOV PETERZIL

§8. Proof of PC for arbitrary deﬁnably compact G. We assume in this section that G is a deﬁnably compact group in an o-minimal expansion of a real closed ﬁeld. Here are some preliminary facts about noncommutative deﬁnably compact groups: As shown in [31], G/Z(G) is semisimple, namely has no inﬁnite deﬁnable abelian normal subgroup. If we let N = Z(G)0 then the same is true for G/N . By [29], G/N can be written as an almost direct product of deﬁnably almost simple groups and every deﬁnably simple group is deﬁnably isomorphic to a semialgebraic linear group deﬁned over the real algebraic numbers. In particular, deﬁnably simple groups have very good reduction. The proof of PC for groups with very good reduction is partly contained in the Introduction (see [28] for more details). It easily follows that PC holds for deﬁnably compact almost simple groups and therefore also for an almost direct product of such groups. Therefore, PC holds for both N (Theorem 7.1) and for G/N . We thus have: dimM (G) = dimM (G/N ) + dimM (N ) = dimLie ((G/N )/(G/N )00 ) + dimLie (N/N 00 ). We also have dimLie (G/G 00 ) = dimLie (G/G 00 N ) + dimLie (G 00 N/G 00 ). It is easy to verify that G 00 N/N = (G/N )00 . Hence, (G/N )/(G/N )00 G/G 00 N. In order to show that dimM (G) = dimLie (G/G 00 ) it is therefore suﬃcient to prove: G 00 N/G 00 N/N 00 . The group on the left is isomorphic to N/(G 00 ∩ N ), hence in order to prove PC we are left to prove: Lemma 8.1. If G is deﬁnably compact then N 00 = G 00 ∩ N . The fact that N 00 ⊆ (G 00 ∩ N ) follows from the fact that the group on the right has bounded index in N . However, in order to prove the opposite inclusion (which fails for arbitrary groups, even with NIP) we need to take one more de’tour, through the general theory of generic sets. §9. Some theory of generic sets III. In this section we make no assumption on the group G unless otherwise stated.

PILLAY’S CONJECTURE

195

Deﬁnition 9.1. The theory of G is said to have ﬁnitely satisﬁable generics1 (in short fsg) if there exists a complete type p over M such that if φ(x) ∈ p then: (i) φ(G) is left generic. (ii) There exists a small model M0 ⊆ M such that every left translate of φ(G) intersects M0 . Our goal is to show that every deﬁnably compact group in an o-minimal structure has fsg. This is useful because of the following properties: Fact 9.2. Assume that T = Th(G) has fsg. Then (i) There exists a small M0 ⊆ M such that every left generic set and every right generic set intersect M0 . (ii) Given X ⊆ G deﬁnable, X is left-generic if and only if it is right generic. (iii) The deﬁnable non generic

sets in G form an ideal. (iv) G 00 exists and G 00 = {Stabng (X ) : X ∈ Def(G)}. Proof. Assume that p and M0 witness fsg. (i) If X is a left generic set then there are g1 , . . . , gk ∈ G such that the formula x = x is equivalent to the ﬁnite disjunction of the formulas x ∈ gi X . Hence, there is gi such that “x ∈ gi X ” is in p. By assumption on p, X ∩ M0 = ∅. Consider the type p(x −1 ). Because x → x −1 is a ∅-deﬁnable bijection of G it easily follows that if φ(x) ∈ p(x −1 ) then Y = φ(G) is right generic and every right translate of Y intersects M0 . As above, it follows that every right generic set intersects M0 . (ii) Assume X is not a left generic set. By Fact 5.6, there exists a right translate of X which does not intersect M0 , hence by (i), X is not right generic as well. (iii) Since p is a complete generic type it must contain the complement of every nongeneric set. (iv) For the existence of G 00 , see [18], Corollary 4.3. Now ﬁx a small model M0 witnessing (i). Given X ⊆ G deﬁnable, let M1 be a small model containing M0 over which X is deﬁnable. If g ≡M1 h then gX ∩ M1 = hX ∩ M1 and hence (gX hX ) ∩ M0 = ∅. By (i), gX hX is nongeneric. Thus, every coset of Stabng (X ) contains all the realizations of some complete type over M1 . In particular, in a (still small) model where every complete type over M1 is realized, there is a representative for every coset of Stabng (X ), so Stabng (X ) has bounded index, and therefore it contains G 00 . For the opposite inclusion, since G 00 has bounded index it can be obtained as the intersection of deﬁnable generic sets (Fact 5.5). If g belongs to Stabng (X ) for every such X then it must belong to G 00 (otherwise, by compactness, there is X containing G 00 such that gX ∩ X = ∅, which implies that g∈ / Stabng (X ). 1 For

simplicity, we slightly modiﬁed the deﬁnition from [18].

196

YA’ACOV PETERZIL

Lemma 9.3. Assume that N is a deﬁnable normal subgroup of G. If N and G/N have fsg then G has fsg. Proof. See Proposition 4.5 in [18]. We return to the o-minimal setting. Lemma 9.4. If G is deﬁnably compact and abelian in an o-minimal structure then G has fsg. Proof. Since we do have complete generic types in abelian groups (see Fact 5.8), it is suﬃcient to show that there is a small M0 such that every generic set intersects M0 . Let M0 be a small model such that every ≈ng -class in Def(G) has a representative deﬁnable over M0 (recall, Lemma 6.4, that ≈ng is bounded because G has NIP). Given X ⊆ G generic, there exists X1 ⊆ X such that X1 is still generic and Cl (X1 ) ⊆ X . Indeed, the following argument for that fact was suggested by the UIUC Logic seminar (it assumes that M expands an ordered group but this is unnecessary, as the argument in [18] shows): G=

k

gi X.

i=1

By Fact 5.4, Int(X ) is also generic. For every > 0 let X = {g ∈ X : d (g, Fr(X )) > } (the notion of d (g, Fr(X )), the distance of g from the frontier of X , assumes the presence of an underlying group). We have, X . X = >0

It is now suﬃcient to take realizing the complete type p(x) of the inﬁnitesimals right of zero. So, G=

k

gi X .

p i=1

We obtained a deﬁnable open covering of G parameterized by a complete type. By the equivalent deﬁnition to deﬁnable compactness, Fact 2.3, there is a ﬁnite subcover, which easily implies that some X is generic. If we let X1 = X then Cl (X1 ) ⊆ X . We may therefore assume that X is closed. Let Y ⊆ G be a set deﬁnable over M0 such that Y X is nongeneric (the existence of Y follows from our assumption on M0 ). Because X is generic so is Y . Again, by Fact 5.4, we may assume that Y is closed, so both X and Y are deﬁnably compact. We will show that (Y ∩ X ) ∩ M0 = ∅ and in particular X ∩ M0 = ∅. Notice that both X and Y are deﬁnably compact.

PILLAY’S CONJECTURE

197

By Theorem 5.7, if X ∩ Y ∩ M0 = ∅ then there are ﬁnitely many M0 conjugates of X ∩ Y whose intersection is empty. Because Y is M0 -deﬁnable there are X1 , . . . , Xk all M0 -conjugates of X such that k

Xi ∩ Y = ∅.

i=1

Since Y is generic this implies that for some Xi we must have Y \ Xi generic. Contradiction to Y X being non-generic. Lemma 9.5. If H is deﬁnably compact and semisimple in an o-minimal structure then H has fsg. The proof of this lemma is based on the almost-decomposition into deﬁnably almost simple groups. The deﬁnably simple case is handled in [28] using measure theoretic arguments based on [7] and [1]. Using Lemmas 9.4, 9.5, and 9.3, we can conclude: Theorem 9.6. Every deﬁnably compact group in an o-minimal structure has fsg. The above theorem, together with Fact 9.2, implies that the set of left (hence also right) generics in G form an ideal, and that for any deﬁnable set X , Stabng (X ) is a type-deﬁnable group of bounded index. Finally (and this is the main fact which forced us to take this de‘tour through the notion of “fsg”), the group G 00 is the intersection of all stabilizers of deﬁnable subsets of G. §10. Completing the proof of PC. We can now return to the missing ingredient in the proof of PC, namely the proof of Lemma 8.1. We need to show that G 00 ∩ N = N 00 , where N is a deﬁnably connected normal central subgroup. By Corollary 4.2, it is suﬃcient to prove that G 00 ∩N is torsion-free. Given n ∈ N, let Tn = Torn (N ) and X ⊆ N be a deﬁnable set such that g → g n gives a bijection of X and N . By Deﬁnable Choice, there is D ⊆ G which intersects every coset of N exactly once. It is now easy to verify that G is the ﬁnite disjoint union of the translates of DX by the elements of Tn . In particular, DX is generic and Tn ∩ Stabng (DX ) = {e}. Because this is true for every n, we have Stabng (DX ) ∩ Tor(G) = ∅. By the fsg property, G 00 ⊆ Stabng (DX ), therefore G 00 ∩ Tor(N ) = {e}. We thus proved that G 00 ∩ N = N 00 , completing the proof of PC (see the argument preceding Lemma 8.1). 10.1. Deﬁning measure on G. As a result of the work on Pillay’s Conjecture, the following theorem was established in [18].

198

YA’ACOV PETERZIL

Theorem 10.1. If G is deﬁnably compact in an o-minimal structure then it admits a left-invariant Keisler measure on the deﬁnable subsets of G. For a deﬁnable X ⊆ G, we have (X ) = 0 if and only if X is non-generic. Proof. As we already pointed out, the existence of such measure is immediate when G is abelian. In the general case, we ﬁrst note that G/G 00 , as a compact Lie group, admits a left-invariant ﬁnitely additive probability measure on a boolean algebra of sets containing all Borel measurable sets- the Haar measure m. We ﬁrst ﬁx a complete generic type p(x) over G. Given a deﬁnable set X , we consider the set Xˆ = {gG 00 ∈ G/G 00 : p |= “x ∈ gX }. (note that X is well deﬁned. Namely, if gh −1 ∈ G 00 then in particular, gX hX is non-generic and therefore not in p. It follows that gX ∈ p if and only if hX ∈ p). The main part of the proof is to show that Xˆ is a Borel set in G/G 00 (see Proposition 6.2 in [18]). We then deﬁne p (X ) = m(Xˆ ). Clearly, p is left invariant, and it is easy to verify that it is also ﬁnitely additive (if X1 ∩ X2 = ∅ then Xˆ1 ∩ Xˆ2 = ∅). Finally, if X is generic then ﬁnite additivity implies that p (X ) > 0 and if X is non-generic then Xˆ = ∅ and therefore p (X ) = 0. §11. Related work and some open questions. This section has gone through substantial changes in the last stages of writing. As will be explained below, most of the open questions listed here were solved in a recent paper by Hrushovski and Pillay, [20]. 11.1. Omitting the real closed ﬁeld assumption. As was pointed out early on, the only remaining obstacle for proving PC without the assumption that M expands a real closed ﬁeld is the lack of an analogue to Theorem 7.2 on the number of torsion points, without the ﬁeld assumption. Such a theorem was proved by Eleftheriou and Starchenko [16] when M was assumed to be an ordered division ring over an ordered vector space and hence PC holds in this case as well. Actually, a very clear description of deﬁnable groups in this setting is given in the paper, out of which the number of torsion points is easily read. In order to prove the torsion points result under weaker assumptions it seems important to develop similar topological tools to the ones originally used, but this time without the triangulation theorem. Indeed, Sheaf Cohomology in expansions of ordered groups has been the subject of several papers of Edmundo, Jones and Peatﬁeld (see [11] and [12]) and of Beraducci and Fornasiero (see [4]). In [14], Edmundo and Terzo prove Pillay’s conjecture

PILLAY’S CONJECTURE

199

under a relatively weak assumption on M but with an additional assumption of “orientability” on the group G. In a recent result, [27], I was able to prove the question about the torsion points and hence Pillay’s Conjecture, in o-minimal expansions of ordered groups, follows. The questions formulated below were written prior to the publication of the recent pre-print by Hrushovski and Pillay [20]. As I will eventually point out, most of these questions are now solved by that paper, either explicitly or implicitly. I leave them here because I ﬁnd that their discussion could still be of some interest. 11.2. Uniform deﬁnability of G 00 . An important feature of the basic example of Pillay’s conjecture (where we start with a compact real Lie group and view it in an elementary extension) is the fact that the type deﬁning G 00 is given by a single formula, with varying parameters. Namely, G 00 = {g ∈ G : |g| < a : a ∈ R}. Consider the structure Gind whose universe is G/G 00 , with a function symbol for the group operation and a predicate for every set of the form (X ), for X ⊆ G n deﬁnable in the o-minimal structure M. In [18] we showed, using a theorem of Baysalov and Poizat, that if G = [0, 1)n , + mod 1 (in an ominimal expansion of an ordered divisible abelian group) then structure Gind is deﬁnable in an o-minimal structure over the reals. Later, in [22], Marikova reproved this result without referring to [1], and provided a much ﬁner analysis of the deﬁnable sets in this structure. The uniformity in parameters plays an important role in both works. For more recent work of Marikova and v.d. Dries see [23], [39] Conjecture. If G is deﬁnably compact then there is a formula φ(x, y), where x varies over elements of G, and a set of parameters A, such that φ(g, a) . G 00 = g ∈ G : a∈A

Conjecture. The structure on the compact Lie group Gind is deﬁnable in some o-minimal structure over the real numbers. Related to the above conjecture is the following: Question. What is the structure which G induces on Tor(G)? In particular, what subsets of Tor(G) are of the form X ∩ Tor(G) for a deﬁnable subset of G? Note that when G is abelian its torsion group can be realized as a deﬁnable set in the o-minimal structure Q, +, 0. Proof. Existence and uniqueness of H is clear. Assume there are B, ε as stated. Then K := B \ int{F = 0} is compact and G does not have zeroes on K . Let c ∈ R, such that c ≥ ε1 and c1 ≤ |G|K |. Then for every x ∈ Rn we have |H (x)| ≤ c · |F (x)|: this holds true if F (x) = 0, since F = H · G and H vanishes on {G = 0}. If F (x) = 0, then x ∈ B or x ∈ B \ int{F = 0} = K . In both cases we get the assertion by the choice of c. 3.3. Corollary. Let A be bounded super real closed. Let r ∈ R, F1 , F2 ∈ C pol (Rn ) and a1 , . . . , an ∈ A be such that |ai | ≤ r (1 ≤ i ≤ n) and such that

BOUNDED SUPER REAL CLOSED RINGS

225

F1 (x) = F2 (x) for all x in an open set containing [−r, r]n ⊆ Rn . Then F1,A (a1 , . . . , an ) = F2,A (a1 , . . . , an ). n Proof. Deﬁne G : Rn −→ R by G(x) = i=1 sup{0, |xi | − ri }. Then G ∈ C (Rn ) and {G = 0} = [−r, r]n ⊆ int{F1 − F2 = 0} by assumption. By 3.2, there is some H ∈ C (Rn ) with F1 − F2 = H · G and since G ≥ 1 outside [−r − 1, r + 1]n we know that |H | ≤ c · |F1 − F2 | for some c ∈ R. Since F1 , F2 are polynomially bounded, also H ∈ C pol (Rn ). Thus F1,A (a1 , . . . , an ) − F2,A (a1 , . . . , an ) = HA (a1 , . . . , an ) · GA (a1 , . . . , an ). Since n|ai | ≤ r for each i and A is a real closed ring we know that GA (a1 , . . . , an ) = i=1 sup{0, |ai | − ri } = 0, which implies the corollary. 3.4. Proposition and Deﬁnition. Let A be a bounded super real closed ring. The holomorphy ring Hol A is a bounded super real closed subring of A and there is a unique super real closed ring structure on Hol A, which expands the bounded super real closed ring structure. For F ∈ C (Rn ) and a1 , . . . , an ∈ Hol(A) we have (†)

FHol A (a1 , . . . , an ) = GHol A (a1 , . . . , an )

whenever G ∈ C pol (Rn ) is such that for some r ∈ N with |ai | ≤ r we have F (x) = G(x) (x ∈ Rn , |x| ≤ r + 1). Proof. Hol A is a bounded super real closed subring of A, since for all a1 , . . . , an ∈ Hol A and each F ∈ C pol (Rn ), there are r ∈ R with |ai | ≤ r and a bounded F ∗ ∈ C ∗ (Rn ) such that F (x) = F ∗ (x) (|x| ≤ r + 1); hence by 3.3, FA (a1 , . . . , an ) = FA∗ (a1 , . . . , an ) ∈ Hol A. By 3.3, we may use (†) to deﬁne an LΥ -structure on Hol(A) which by deﬁnition expands the bounded super real closed ring structure on Hol A. It is straightforward (using 3.3) to check that this deﬁnes the unique super real closed ring structure on Hol A which expands the bounded super real closed ring structure. pol n 3.5. Theorem. Let F ∈ C (R ) and P(T ) ∈ R[T ], T = (T1 , . . . , Tn ) of total degree d with |F | ≤ |P| on Rn . Then there is a unique continuous function G ∈ C (Rn × R) with (∗)

¯ y) ∈ Rn × R). F (x1 , . . . , xn ) · y d +1 = G(x1 · y, . . . , xn · y, y) ((x,

Moreover G is polynomially bounded. Proof. Existence is given by [Tr2, (7.2)](ii). Uniqueness holds, since ¯ y) ∈ Rn × R with G(x1 , . . . , xn , y) is uniquely determined by (∗) for all (x, y = 0. Moreover the proof of [Tr2, (7.2)](ii) shows that G is again polynomially bounded (by a polynomial of total degree d . 3.6. Theorem. Let A be a bounded super real closed ring and let 1 ∈ S ⊆ A be multiplicatively closed. Then there is a unique expansion of the localization

226

MARCUS TRESSL

S −1 A to a bounded super real closed ring such that the localization map A −→ S −1 A is a bounded super homomorphism. The operation of F ∈ C pol (Rn ) on (S −1 A)n is given as follows: Pick d ∈ N0 such that F is bounded by a polynomial of total degree d and take a polynomially bounded continuous function G ∈ C (Rn × R) with F (x1 , . . . , xn ) · y d +1 = G(x1 · y, . . . , xn · y, y) ((x, ¯ y) ∈ Rn × R). Such functions exist by 3.5. Then for f1 , . . . , fn ∈ A and g ∈ S fn GA (f1 , . . . , fn , g) f1 ,..., := ∈ S −1 A. FS −1 A g g g d +1 Proof. The proof is parallel to the proof of the localization theorem [Tr2, (7.4)], using 3.5 instead of [Tr2, (7.2)](i). 3.7. Corollary. Let ϕ : A −→ B be a super homomorphism between bounded super real closed rings and let 1 ∈ S ⊆ A be multiplicatively closed such that ϕ(S) ⊆ B × . Then the natural map S −1 A −→ B is a super homomorphism, too. Proof. This follows immediately from the explicit deﬁnition of the bounded super real closed structure on S −1 · A in 3.6. §4. The super real closed hull. For a bounded super real closed ring A, we shall now deﬁne the smallest super real closed ring containing A as a bounded super real closed subring. 4.1. Theorem and Deﬁnition. Let A be a bounded super real closed ring. Let Aˆ = S −1 · Hol A, where S is the closure of A× ∩ Hol A under multiplication and Υ (recall: this means “closed under all the functions sHol A , s ∈ Υ”); here we consider Hol(A) equipped with the super real closed ring structure deﬁned in 3.4. Then there is a unique LΥ -structure on Aˆ such that Aˆ is a super real closed ring having A as a bounded super real closed subring. Aˆ is called the super real closed hull of A. Proof. Firstly, as A× ∩ Hol A ⊆ S we have A = (A× ∩ Hol A)−1 · Hol A ⊆ −1 ˆ By 3.4, Hol(A) is a bounded super real closed subring of S · Hol A = A. A and there is a unique expansion of this structure to a super real closed ring. By deﬁnition, S is closed under multiplication and Υ. By 3.1, there is a unique LΥ -structure on Aˆ such that Aˆ is a super real closed ring having Hol A as a super real closed subring. Since Aˆ is also the localization of A at S, 3.6 implies ˆ that A is a bounded super real closed subring of A. ˆ It remains to show that A with the LΥ -structure deﬁned above is the unique super real closed ring structure on Aˆ having A as a bounded super real closed subring. However, any other super real closed ring B expanding the pure

BOUNDED SUPER REAL CLOSED RINGS

227

ring Aˆ having A as a bounded super real closed subring, has Hol A as a super real closed subring (cf. [Tr2, (9.2)](i)) and the underlying bounded super real closed ring structure is the one induced from A. By 3.4, the super real closed ring structures of B and Aˆ induced on Hol A are equal. From the uniqueness ˆ property in 3.1 we know that B is the super real closed ring A. 4.2. Corollary. Let F, G ∈ C pol (Rn ). ¯ = 0 → G(x) ¯ = 0. (i) If {F = 0} ⊆ {G = 0}, then TΥpol ∀x¯ F (x) ¯ ≥ 0 → G(x) ¯ ≥ 0. (ii) If {F ≥ 0} ⊆ {G ≥ 0}, then TΥpol ∀x¯ F (x) Proof. (i). Let A |= TΥpol . By [Tr2, (5.5)](iv) the super real closed ring Aˆ is a model of ¯ = 0 → G(x) ¯ = 0. ∀x¯ F (x) Since A is a bounded super real closed subring of Aˆ (by 4.1), also A is a model of this sentence. (ii) follows from (i), since in every real closed ring A, the formula x ≥ 0 is equivalent to pA (x) = 0, where p : R −→ R is the inﬁmum of the identity function and the constant function 0. 4.3. Lemma. Let A be a bounded super real closed subring of the super real closed ring B. There is a unique A-algebra homomorphism Aˆ −→ B and this homomorphism is an embedding of super real closed rings. Proof. We have S0 := A× ∩ Hol A ⊆ T := B × ∩ Hol B. Since B is super real closed, T is closed under Υ: this follows from [Tr2, (6.12)], which says that all maximal ideal of B are Υ-radical. Since Hol A is a super real closed subring of Hol B by 3.4, T ∩ Hol A is Υ-closed as well. Thus the closure S of S0 under Υ and multiplication is contained in T , too. Hence we get a unique A-algebra homomorphism ϕ : Aˆ = S −1 · Hol(A) −→ T −1 · Hol(B) = B and this map is injective. It remains to show that ϕ is a super homomorphism. This follows immediately from the deﬁnition of the LΥ -structure on both rings in 3.1. 4.4. Corollary. If A is a super real closed ring, then Aˆ (deﬁned for the underlying bounded super real closed ring) is equal to A. In particular, the LΥ -structure of A is uniquely determined by the LΥpol -structure. 4.5. Corollary. Let B be a super real closed ring and let A be a bounded super real closed subring of A. Then B ∼ =A Aˆ as (bounded ) super real closed rings if and only if B is generated by A as a super real closed ring. Proof. Let C ⊆ Aˆ be the super real closed subring generated by A. By 4.3 there is a super real A-algebra monomorphism Aˆ −→ C . Composing this ˆ map with the inclusion C −→ Aˆ and using uniqueness shows that C = A. ˆ Hence A is generated by A as a super real closed ring.

228

MARCUS TRESSL

Conversely suppose B is generated by A as a super real closed ring. By 4.3, we may view Aˆ as a super real closed subring of B. Since B is generated by A ˆ we get B = A. 4.6. Proposition. Let A be a bounded super real closed ring. Then A is ˆ in other words Aˆ is a subring of the convex closure B of A. There convex in A, is a unique super real closed ring-structure on B extending the bounded super real closed ring structure on A. In particular, every bounded super real closed ring which is convexly closed (e.g. a ﬁeld ) has a unique expansion to a super real closed ring. Proof. Since Hol(A) is convex in A, the convex closure B of Hol A contains A. By [Tr2, (11.2)](iii) we know that B is the localization of Hol A at the set T of all non zero-divisors t of Hol A with the property that Hol A is convex in (Hol A)t . It follows A× ∩ Hol A ⊆ T . Since T is closed and closed under Υ by [Tr2, (11.11)], the closure S of A× ∩ Hol A is contained in T . Hence ˆ Aˆ = S −1 · Hol(A) ⊆ T −1 · Hol(A) = B, in other words A is convex in A. By [Tr2, (11.12)], there is a (unique) expansion of B to a super real closed ring having Aˆ as super real closed subring. Since B is a localization of Aˆ we get the uniqueness statement of the proposition from the uniqueness statement in 3.6 together with 4.4. 4.7. Corollary. Let A be a bounded super real closed subring of a super real closed ring B. Then A is convex in the super real closed ring generated by A in B. Proof. By 4.5 and 4.6. §5. Super real ideals. 5.1. Deﬁnition. An ideal I of a bounded super real closed ring A is called super real if sA (I ) ⊆ I for every s ∈ Υpol . Observe that in this case I is a radical ideal, in particular I is convex and satisﬁes a ∈ I ⇐⇒ |a| ∈ I (a ∈ A). Certainly,√ every ideal I of A is contained in a smallest super real ideal of A, denoted by Υ I . If A is a super real closed ring, then by [Tr2, (6.10)], the super real ideals are precisely the Υ-radical ideals (clearly Υpol is a set of generalized root functions as deﬁned in [Tr2, (3.2)]). 5.2. Examples. Let A be a bounded super real closed ring. (i) If F ∈ C pol (R) is strictly positive everywhere, then in general FA (a) is not a unit for every unit a ∈ A. For example if A is the bounded super real closed ring C pol (R), F = exp(−x 2 ) and a = 1 + x 2 ∈ A. (ii) If a ∈ A is a unit, then in general, there is some s ∈ Υpol , which is bounded away from 0 outside [−1, 1] such that sA (a) is not a unit. For

BOUNDED SUPER REAL CLOSED RINGS

229

1 ) example if A is the bounded super real closed ring C pol (R), s = exp(− |x| 1 and a = 1+x 2 ∈ A. Hence in this example, the ideal I = (s(a)) of A is proper, but the super real radical of I is not proper. In particular, maximal ideals of bounded super real closed rings are not super real in general. The example also shows that this is not resolved if we replace Υpol by the set of all s ∈ Υpol , which are bounded away from 0 outside a neighborhood of 0: or to replace Υpol by the set of all bounded s ∈ Υ such that s does not have zeroes diﬀerent from 0 in the Stone-Cech compactiﬁcation R of R.

5.3. Remark. If A ⊆ B is an extension of rings and I is an ideal of A, then I · B denotes the ideal generated by I in B. Recall that for a convex subring A of a real closed ring B and every radical ideal I of A we have I · B = {a · b | a ∈ I, b ∈ B} and this ideal is again radical. Our ﬁrst goal in this section is to show that 5.3 remains valid in the bounded super real closed context. That is, whenever A ⊆ B is a convex extension of bounded super real closed rings and I is a super real ideal of A, then I · B is a super real ideal of B (cf. 5.7). In order to prove this we show that for every s ∈ Υpol , there are t ∈ Υpol and F ∈ C pol (R2 ) with s(x · y) = t(x) · F (x, y). This is achieved in 5.6 below. First a preparational lemma: First two preparational lemmas from elementary analysis: 5.4. Lemma. Let A ⊆ R2 be compact with projection [a, b] onto the ﬁrst coordinate. Let C be the convex hull of A. Then C is again compact and the function f : [a, b] −→ R deﬁned by f(x) = max Cx is continuous, concave (i.e. f(x + (1 − )y) ≥ f(x) + (1 − )f(y) for all 0 ≤ ≤ 1) and satisﬁes f(a) = max Aa , f(b) = max Ab . Here Cx denotes the set {y ∈ R | (x, y) ∈ C } and similarly for Aa , Ab . Moreover, if A is the graph of a strictly increasing function [a, b] −→ R, then also f is strictly increasing.

Proof. This is clear.

5.5. Lemma. Let sn ∈ Υ (n ∈ N). Then there is some t ∈ Υ, 0 ≤ t ≤ 1, symmetric (i.e. t(−x) = t(x)), non-decreasing and concave in [0, ∞) such that for every n ∈ N there is some > 0 with t(x) ≥ |sn (x)| (|x| < ). Proof. We may assume that all sn are symmetric, non-decreasing on [0, ∞) and 0 ≤ sn ≤ 12 . Deﬁne 1 |x| 1 , + sup min , sn (x) (x) = min 2 2 n n

230

MARCUS TRESSL

which is continuous on R, strictly increasing in [0, 1] with values in [0, 1], (0) = 0 such that for every n, there exists > 0 with (x) ≥ sn (x) for 0 ≤ x ≤ . We deﬁne a function t : [0, 1] −→ [0, 1] as follows: Let C be the convex hull of the graph of [0, 1] and let t(x) := sup Cx (0 ≤ x ≤ 1), where Cx = {y ∈ R | (x, y) ∈ C }. By 5.4, t is a strictly increasing and concave homeomorphism [0, 1] −→ [0, (1)]. We extend t to R via t(x) = t(1) if x ≥ 1 and t(x) = t(−x) if x < 0. Then t is symmetric, 0 ≤ t ≤ 1 and it is clear that t is still concave in [0, ∞). Since t(x) ≥ (x) for all x ∈ [0, 1] it is clear that for all n ∈ N, there is some > 0 with t(x) ≥ sn (x) (|x| < ). 5.6. Proposition. Let s ∈ Υ. There are t ∈ Υ with 0 ≤ t ≤ 1, c ∈ R and F ∈ C (R2 ) such that s(x · y) = t(x) · F (x, y) and |F (x, y)| ≤ c · 1 + (1 + |y|) · |s(x · y)| ((x, y) ∈ R2 ) Proof. Let s0 (x) = x and for n > 0, sn (x) := max|y|≤n n · |s(y · x)|. Then sn ∈ Υ and from 5.5 we get some t ∈ Υ, symmetric with 0 ≤ t ≤ 1, nondecreasing and concave in [0, ∞) such that for every n ∈ N0 there is some > 0 with t(x) ≥ |sn (x)| (|x| < ). By deﬁnition of sn for n ≥ 1 this means (∗)

t(x) ≥ n · |s(yx)| (|x| < , |y| ≤ n).

We ﬁrst show that the function s(x·y) , deﬁned on (R\{0})×R has a continuous t(x) extension F through 0 on R × R: | < n1 for all Pick b ∈ R. For n ∈ N we have to ﬁnd some > 0 with | s(x·y) t(x) x ∈ (−, ), x = 0 and all y with |b − y| < . Enlarge n if necessary such that |b| < n and take > 0 with |b| + < n such that (∗) holds. Let 0 < |x| < and |b − y| < . Then |y| < |b| + < n. Thus |s(xy)| ≤ n1 t(x), as desired. It remains to ﬁnd c ∈ R such that for all (x, y) ∈ R2 , x = 0 we have s(x · y) (†) t(x) ≤ c · (1 + (1 + |y|) · |s(x · y)|). By choice of t there is some > 0 such that t(x) ≥ |s(x)| and t(x) ≥ |x| for all x with |x| < . It is enough to ﬁnd an element c satisfying (†) separately on each of the following four subsets of R2 , covering R2 : Case 1. |x| ≥ .

BOUNDED SUPER REAL CLOSED RINGS

231

Then t(x) = t(|x|) ≥ t() > 0 since t is symmetric and increasing in [0, ∞). 1 | ≤ | s(x·y) | and we may choose c := t() . Hence | s(x·y) t(x) t() Case 2. |x| < and |y| ≤ 1. As F is continuous we may choose c as the maximum of |F | on the rectangle [−, ] × [−1, 1]. Case 3. |x| < and |y| ≥ 1 and |x · y| ≥ . Then by the choice of we have t(x) = t(|x|) ≥ |x|, hence | s(x·y) | ≤ t(x) s(x·y) | s(x·y) x | ≤ |y · |, since 1 choose c = .

1 |x|

≤ | y | by assumption in case 3. Hence we may

Case 4. |x| < and |y| ≥ 1 and |x · y| < . 1 1 ≤ 1 we have t(x) = t(|x|) = t( |y| · |x · Since t is concave in [0, ∞) and |y| 1 1 y|) ≥ |y| · t(|x · y|) = |y| · t(x · y). Since |x · y| < we have t(x · y) ≥ |s(x · y)| by the choice of . Hence s(x · y) t(x · y) t(x) ≤ |y| · t(x · y) = |y| and we may choose c = 1.

5.7. Proposition. Let A be a convex subring of a bounded super real closed ring B. If I is a super real ideal of A then I · B is super real, too. Proof. For a ∈ I , b ∈ B and s ∈ Υpol we have to show that sB (a ·b) ∈ I ·B. By 5.6 there are t ∈ Υpol , c > 0 and F ∈ C (R2 ) with s(x · y) = t(x) · F (x, y) ((x, y) ∈ R2 ) such that |F (x, y)| ≤ c · (1 + (1 + |y|) · |s(x · y)|) everywhere. Since s is polynomially bounded also F is polynomially bounded. Hence sB (a · b) = tB (a) · FB (a, b). Since tB (a) = tA (a) ∈ I we get the claim. If A is a super real closed ring and I is an ideal of A, then there is a largest super real ideal I Υ of A contained in I and I Υ = {a ∈ I | sA (a) ∈ I for all s ∈ Υ}. (cf. [Tr2, (6.7)]). With the aid of 5.7, this can be extended to bounded super real closed rings: 5.8. Proposition and Deﬁnition. Let A be a bounded super real closed ring. If I is an ideal of A, then there is a largest super real ideal I Υ contained in I . We have I Υ = {a ∈ I | sA (a) ∈ I for all s ∈ Υpol } = (I ∩ Hol A)Υ · A. Proof. Let J := (I ∩ Hol A)Υ . By 5.7 we know that J · A is super real. Moreover it is clear that every super real ideal of A contained in I has to be contained in K := {a ∈ I | sA (a) ∈ I for all s ∈ Υpol }. In particular J · A ⊆ K and it remains to show that K ⊆ J · A.

232

MARCUS TRESSL

1 a a Pick a ∈ K . Since 1+a 2 , 1+a 2 ∈ Hol A we have 1+a 2 ∈ I ∩ Hol A and it a a Υ remains to show that 1+a 2 ∈ (I ∩ Hol A) . It suﬃces to show sA (| 1+a 2 |) ∈ I for every strictly increasing s ∈ Υ and indeed by [Tr2, (6.7)] it suﬃces to take √ √ a a | ≤ |a| we have s (| |) ≤ s (|a|) ∈ I by our s ∈ Υpol . Since | 1+a 2 A 1+a 2 A 1 choice √ of a in K . Now the aconvexity condition fora real closed rings implies that s A (|a|) divides sA (| 1+a 2 |) in A. Hence sA (| 1+a 2 |) ∈ I as desired. 5.9. Theorem. Let I be an ideal of a bounded super real closed ring A. Then √ √ Υ Υ Υ Υ I · Aˆ = I · Aˆ and I = I · Aˆ ∩ A. √ √ Υ Υ Proof. The inclusionΥ I · Aˆ ⊆ I · Aˆ follows from Υ I ⊆ I · Aˆ and √ √ Υ Υ I · Aˆ ⊇ I · Aˆ holds, since by 5.7, Υ I · Aˆ is super real. the inclusion √ √ Υ Υ ˆ ∩ A contains I and it remains to show that Clearly ( I · A) √ √ Υ ˆ ∩ A ⊆ Υ I. ( I · A) √ ˆ ∩ A. In order to show b ∈ I We may assume that I = Υ I . Take b ∈ (I · A) 2 we may replace b by b , hence we may assume that b ≥ 0. Since 1 + b 2 is a b b 2 ˆ unit in A we have 1+b 2 ∈ (I · A) ∩ A. Since b = 1+b 2 · (1 + b ) we may replace b ˆ b with 1+b 2 and we may assume that 0 ≤ b ≤ 1. Since b ∈ I · A, there are ˆ a ∈ I and c ∈ A with b = a · c. As b ≥ 0, b = |b| = |c| · |a| and we may assume that a, c ≥ 0, too (observe that I is radical, hence |a| ∈ I ). By 4.5, Aˆ is generated by A as a super real closed ring. Thus there are F ∈ C (Rn ) and a1 , . . . , an ∈ A with c = FAˆ(a1 , . . . , an ). Pick ϕ : [0, ∞) −→ [1, ∞) continuous and strictly increasing with |F (x)| ¯ ≤ ϕ(|x|) ¯ (x¯ ∈ Rn ). Deﬁne t : R −→ R by ⎧ |y| ⎪ ⎨ if y = 0 1 t(y) = ϕ( y 2 ) ⎪ ⎩ 0 if y = 0.

Using [Tr2, (7.2)](i) with s(x) = x we get t ∈ Υ and some G ∈ C (Rn × R) with (∗)

F (x1 , . . . , xn ) · t(y) = G(x1 y, . . . , xn y, y) on Rn × R.

Since ϕ is strictly increasing and ≥ 1 everywhere it is straightforward to see that t|[0.∞) : [0.∞) −→ [0.∞) is an homeomorphism which is polynomially bounded and whose compositional inverse is polynomially bounded, too. Hence t ∈ Υpol and there is some t1 ∈ Υpol with t ◦ t1 (y) = t1 ◦ t(y) = y for all y ≥ 0. As a ≥ 0 we get a = tA (t1,A (a)) from 4.2(ii). Since I is a super real ideal, also a0 := t1,A (a) ∈ I . From (∗) we then get b = c · a = FAˆ(a1 , . . . , an ) · tAˆ(a0 ) = GAˆ(a1 · a0 , . . . , an · a0 , a0 ). 1 The

convexity condition says: 0 ≤ a ≤ b ⇒ b|a 2 .

BOUNDED SUPER REAL CLOSED RINGS

233

Let H := (G ∨ 0) ∧ 1. Then in R we have ∀(x, ¯ y) : 0 ≤ G(x1 y, . . . , xn y, y) ≤ 1 ⇒ G(x1 y, . . . , xn y, y) = H (x1 y, . . . , xn y, y). Since this sentence is also valid in Aˆ and 0 ≤ b ≤ 1 we get b = HAˆ(a1 · a0 , . . . , an · a0 , a0 ). Since H is bounded it follows b = HA (a1 · a0 , . . . , an · a0 , a0 ). As H (0) = 2 ) for G(0) = 0, there is some s ∈ Υpol with H (z1 , . . . , zn+1 ) ≤ s(z12 + · · · + zn+1 2 all z1 , . . . , zn+1 ∈ R: choose s so that s(t) ≥ max{H (z) | zi ≤ t} (t ≥ 0). It follows 0 ≤ b = HA (a1 · a0 , . . . , an · a0 , a0 ) ≤ sA ((a1 a0 )2 + · · · + (an a0 )2 + 2 a0 ). Since a0 ∈ I and I is super real, we get b ∈ I as desired. Note that in general for a proper ideal I of a bounded super real closed ring ˆ Υ (e.g. if I · Aˆ = A, ˆ cf. 5.2(ii)) A, the ideal I Υ · Aˆ is properly contained in (I · A) 5.10. Scholium. Let A be bounded super real closed ring. An ideal of A is super real if and only if I is the kernel of a bounded super homomorphism A −→ B into a bounded super real closed ring. Proof. If ϕ : A −→ B is such a homomorphism and a ∈ I , then sA (a) ∈ I , since ϕ(sA (a)) = sB (ϕ(a)) = sB (0) = 0. Conversely suppose I is super real. By 5.7, I · Aˆ is super real, too. Together with 5.9 it follows that I ·Aˆ is a super real ideal of Aˆ lying over I . By [Tr2, (6.3)], super real ideals of Aˆ are kernels of super homomorphisms. Hence we can ˆ · Aˆ and we get that I is the kernel of a compose A −→ Aˆ with Aˆ −→ A/I bounded super homomorphism. 5.11. Corollary. Let A be bounded super real closed and let I ⊆ A be a super real ideal. There is a unique LΥpol -structure on A/I such that A/I is a bounded super real closed ring and the residue map A −→ A/I is a bounded super real homomorphism.

and this Moreover, there is a unique A-algebra homomorphism Aˆ −→ A/I ˆ homomorphism is super real with kernel I · A. In particular, there is a unique A-algebra isomorphism of super real closed rings ∼ = ˆ ˆ −→ A/I A/(I · A)

Proof. By 5.7 we know that I · Aˆ is a super real ideal of Aˆ lying over I . Since super real ideals are kernels of super homomorphisms by 5.10, we can ˆ · Aˆ and get that I is the kernel of a bounded compose A −→ Aˆ with Aˆ −→ A/I super homomorphism. The image is A/I and it is clear that the LΥpol -structure on A/I is uniquely determined by saying that the residue map A −→ A/I is a bounded super real homomorphism. ˆ · A, ˆ which is a bounded super We get an embedding of rings A/I −→ A/I

real homomorphism. By 4.3, we may view A/I as a super real closed subring

234

MARCUS TRESSL

ˆ · A. ˆ Since Aˆ is generated by A as a super real closed ring, also A/I ˆ · Aˆ of A/I

ˆ · Aˆ = A/I . Hence is generated by A/I as a super real closed ring, thus A/I

ˆ ˆ There we have a super real homomorphism ϕ : A −→ A/I with kernel I · A. ˆ can only be one such A-algebra homomorphism, since A is the localization of ˆ × ∩ A. A at (A) 5.12. Theorem. If ϕ : A −→ B is a bounded super homomorphism between bounded super real closed rings A and B, then there is a unique extension of ϕ to a ring homomorphism ϕˆ : Aˆ −→ Bˆ and this extension is super real. The functor F from bounded super real closed rings to super real closed rings, which maps A to Aˆ and ϕ to ϕˆ is an idempotent mono-reﬂector. This means: F is left adjoint to the inclusion from the category of super real closed rings into the category of bounded super real closed rings, F ◦ F = F and the adjoint morphism A −→ Aˆ is a monomorphism. Proof. First we prove the assertion about ϕ. Uniqueness again follows ˆ × ∩ A. Existence of ϕˆ from the fact that Aˆ is the localization of A at (A) follows from 5.11 and 4.3. Hence the functor F is well deﬁned. By 4.4, F ◦ F = F , which also shows that F is a reﬂector. F is a mono-reﬂector, since A −→ Aˆ is a monomorphism. ˆ We conclude this section by showing that the reﬂector A → A is also wellbehaved with respect to localization: 5.13. Proposition. Let A be a bounded super real closed ring, let 1 ∈ S ⊆ A be multiplicatively closed and let T be the closure of S in Aˆ under multiplication and Υ. Recall from 3.1 that there is a unique super real closed ring structure on T −1 · Aˆ such that the localization map Aˆ −→ T −1 · Aˆ is a super homomorphism. −1 · A induced by the localization map The natural morphism ϕˆ : Aˆ −→ S −1 · A)× and the induced map ϕ : A −→ S −1 · A, sends T into (S −1 · A T −1 · Aˆ −→ S

is an A-algebra isomorphism of super real closed rings. Proof. Since ϕ(S) consists of units of S −1 · A also ϕ(S) ˆ consists of units −1 of S · A. Since T is the closure of S under multiplication and Υ, ϕ(T ˆ ) is the closure −1 · A is super real closed, every of ϕ(S) ˆ under multiplication and Υ. Since S −1 · A is super real (cf. [Tr2, (6.12)]), hence for every for maximal ideal of S −1 · A, b is a unit in S −1 · A if and only if every s ∈ Υ and each element b ∈ S −1 −1 · A)× . s(b) is a unit in S · A. This proves that indeed ϕ(T ˆ ) ⊆ (S −1 · A is an isomorIn order to show that the induced map T −1 · Aˆ −→ S phism it now suﬃces to verify the universal condition deﬁning T −1 · Aˆ in the

BOUNDED SUPER REAL CLOSED RINGS

235

−1 · A, more precisely for the morcategory of super real closed rings for S −1 · A. Let : Aˆ −→ B be a super homomorphism into a phism Aˆ −→ S super real closed ring B with (T ) ⊆ B × . Then A : A −→ B is a super homomorphism with |A (S) ⊆ B × and by 3.7 there is a unique super homo−1 · A −→ B morphism h : S −1 ·A −→ B such that |A = h ◦ϕ. By 5.12, hˆ : S ˆ is the unique super homomorphism extending h with = h ◦ ϕ. ˆ

§6. The super real core. 6.1. Proposition. Let A0 be a convex subring of the super real closed ring A. Then there is a largest super real closed subring of A that is contained in A0 . Proof. By [Tr2, (9.2)](i), the convex hull of a super real closed subring of A is itself a super real closed subring of A. Hence, by using Zorn, it is enough to show for convex super real closed subrings B, C of A, that the ring D generated by B and C in A is again a super real closed subring of A. By [Tr2, (10.5)] we know that D is a convex subring of A and by [Tr2, (9.2)](i), it is enough to show that D is closed under Υ: Let b1 , . . . , bn ∈ B and c1 , . . . , cn ∈ C . Pick s ∈ Υ. It is enough to show |sA (b1 c1 + · · · + bn cn )| ≤ d for some d ∈ D. We may certainly assume that s is symmetric (i.e. s(−x) = s(x)) and strictly increasing on (0, ∞). The Cauchy-Schwarz inequality implies s(x1 y1 + · · · + xn yn ) ≤ s(|x||y|) for all x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn , where |x|, |y| denote the euclidean norm of x, y respectively. Since s(|x||y|) ≤ s(|x|2 ) + s(|y|2 ) we get s(x1 y1 + · · · + xn yn ) ≤ s(|x|2 ) + s(|y|2 ) on Rn × Rn . Thus sA (b1 c1 + · · · + bn cn ) ≤ sA (b12 + · · · + bn2 ) + sA (c12 + · · · + cn2 ) ∈ D as desired. 6.2. Corollary and Deﬁnition. For any bounded super real closed ring A there is a largest bounded super real closed subring, denoted by AΥ with the Υ = AΥ . We call AΥ the super real core of A. property A ˆ which is Proof. By 6.1, AΥ is the largest super real closed subring of A, contained in A. Observe that AΥ is convex in A, since Hol A ⊆ AΥ . For a proper ideal I of A we know I Υ = (I ∩ Hol A)Υ · A from 5.8. Hence I Υ = (I ∩ AΥ )Υ · A as well. √ √ Υ On the other hand Υ I ∩ AΥ in general properly contains I ∩ AΥ (e.g. if √ Υ I = A). 6.3. Corollary. For any bounded super real closed ring A we have AΥ = {a ∈ A | sAˆ(a) ∈ A for all s ∈ Υ}. Proof. Since AΥ is a super real closed subring of Aˆ we have “⊆”. Conversely take a ∈ A with sAˆ(a) ∈ A for all s ∈ Υ. Let B be the super real closed ˆ Thus B = {F ˆ(a) | F ∈ C (R)}. Certainly every subring generated by a in A. A element of B is bounded in absolute value by some sAˆ(a) for some s ∈ Υ.

236

MARCUS TRESSL

Hence by choice of a, the convex hull C of B in Aˆ is contained in A. C is a super real closed subring of B by [Tr2, (9.2)](i). Hence a ∈ C ⊆ AΥ . 6.4. Observation. If B is a real closed ring and A ⊆ B is a convex subring, then A is a domain if and only if B is a domain, and A is local if and only if B is local (as follows from the Gelfand-Kolmogorov Theorem). In particular for every bounded super real closed ring A we have (i) A is a domain ⇐⇒ Aˆ is a domain ⇐⇒ AΥ is a domain ⇐⇒ Hol A is a domain. (ii) A is local ⇐⇒ Aˆ is local ⇐⇒ AΥ is local ⇐⇒ Hol A is local. 6.5. Examples. Let A := {f ∈ C (R2 ) | f is polynomially bounded in the second coordinate}. Hence f ∈ A if and only if for every x ∈ R, the function f(x, ) : R −→ R is polynomially bounded. Clearly A is a convex subring of C (R2 ), hence A is a bounded super real closed subring of C (R2 ). We have Aˆ = C (R2 ) and AΥ = {f ∈ C (R2 ) | f is bounded in the second coordinate}. Here a super real closed ring properly between C ∗ (R) and C (R): Take A = {f ∈ C (R) | f is bounded on (0, ∞)}. Also note that there are many super real closed ring properly between C ∗ ([0, ∞)) and C ([0, ∞)), e.g. A = {f ∈ C (R) | f is bounded on N} has this property since x · distN (x) ∈ A \ C ∗ (R). The formation of the super real core is functorial: If ϕ : A −→ B is a bounded super homomorphism between bounded super real closed rings, then ϕ|AΥ is a super homomorphism AΥ −→ B Υ : since ϕˆ respects the LΥ structure on Aˆ by 5.12, ϕ(AΥ ) is a super real closed subring of Bˆ contained in B, i.e. ϕ(AΥ ) ⊆ B Υ . Hence the assignment A −→ AΥ is functorial, by sending ϕ to ϕ|AΥ . We shall not make use of this here. Instead, we state another description of the super real core. Since Hol A ⊆ AΥ ⊆ A, there are subsets S of Hol A with AΥ = S −1 ·Hol A. We can compute the largest such set upon input A: 6.6. Proposition. For any bounded super real closed ring A, the largest multiplicatively closed subset S of Hol A satisfying AΥ = S −1 · Hol A is S = {a ∈ Hol A | sHol A (a) ∈ A× for all s ∈ Υ}. Proof. The super real closed subrings of Aˆ contained in A are all of the form T −1 · Hol A, where T ⊆ A× ∩ Hol A. Since AΥ is the largest super real

BOUNDED SUPER REAL CLOSED RINGS

237

closed subring of Aˆ contained in A, the set T := (AΥ )× ∩ Hol A is the largest among all of them. It remains to show T = S. If a ∈ T , then a is a unit in AΥ and since AΥ is super real closed, all elements sAΥ (a) are units of AΥ as well. Since (AΥ )× ⊆ A× we get a ∈ S. Conversely let a ∈ S. The set T0 := {sHol A (a) | s ∈ Υ} is closed under multiplication and Υ (note that s1 , s2 ∈ Υ implies s1 (x) · s2 (x) ∈ Υ and s1 ◦ s2 ∈ Υ). Therefore T0−1 · Hol A has a unique super real closed ring ˆ Since a ∈ S we know T0 ⊆ A× and therefore structure (induced from A). −1 T0 · Hol A ⊆ A. So by the choice of T we obtain T0 ⊆ T . Thus a ∈ T0 ⊆ T . REFERENCES

[Schw] N. Schwartz, The basic theory of real closed spaces, Memoirs of the American Mathematical Society, vol. 77 (1989), no. 397, pp. viii+122. [Tr1] M. Tressl, Computation of the z-radical in C (X ), Advances in Geometry, vol. 6 (2006), no. 1, pp. 139–175. [Tr2] , Super real closed rings, Fundamenta Mathematicae, vol. 194 (2007), no. 2, pp. 121–177. UNIVERSITY OF MANCHESTER SCHOOL OF MATHEMATICS OXFORD ROAD, MANCHESTER M13 9PL, UK

E-mail: [email protected]

ANALYTIC COMBINATORICS OF THE TRANSFINITE: A UNIFYING TAUBERIAN PERSPECTIVE

ANDREAS WEIERMANN

Abstract. From a Tauberian perspective we prove and survey several results about the analytic combinatorics of (transﬁnite) proof-theoretic ordinals. In particular we show how certain theorems of Petrogradsky, Karamata, Kohlbecker, Parameswaran, and Wagner can be used to give a uniﬁed treatment of asymptotics for count functions for ordinals. This uniform approach indicates that (Tauberian theorems for) Laplace transforms provide a general tool to establish connections between additive and multiplicative results and may therefore be seen as a contribution to Problem 12.21 in Burris’s book on number theoretic density and logical limit laws. In the last section we give applications and prove in some detail phase transitions related to Friedman style combinatorial well-orderdness principles for fragments of ﬁrst order Peano arithmetic.

§1. Introduction to the transﬁnite combinatorics of the transﬁnite. Some years ago the author had a discussion with another logician about which ﬁelds in mathematics are surely have no connection with each other. A suggestion was: The theory of transﬁnite ordinal numbers and complex analysis. Of course this seems a safe guess since it is diﬃcult to imagine that Cauchy’s integral formula has something to say about the ordinals below or below ε0 . It has therefore been surprising that those connections exist and give rise to interesting cross-disciplinary applications. Perhaps even more interestingly we apply in this paper results from a paper in Lie-algebra theory to questions in logic. When the author gave a talk about these developments he has moreover been asked whether there are possible connections between ordinals and Tauberian theory, in particular Karamata’s theorem. The questioner was of course convinced that no natural connections exist. But the opposite is true. Karamata’s theorem is of considerable importance for studying Ackermannian functions (when resulting from < -descent recursive functions). Moreover in this article we show how classical Tauberian theorems can be used for studying the provably recursive functions for the fragments of ﬁrst order Peano arithmetic. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

238

ANALYTIC COMBINATORICS OF THE TRANSFINITE

239

Historically the subject analytic combinatorics of the transﬁnite emerged from investigations of the author about Friedman style independence results for PA [35]. For these investigations asymptotic results on the counting of ordinals of a prescribed complexity turned out to be of crucial importance. Quite a few logicians would be content with the insight that these complexities are bounded exponentially but for the purpose of classifying phase ¨ transitions for Godel incompleteness results better bounds are needed. This is exactly the place where complex analysis and Tauberian theorems come into play. Cauchy’s formula provides bounds for the coeﬃcients of the generating function for the objects in question. (Similarly Tauberian theory, for example, via Karamata’s or Ingham’s theorem can be used for related purposes.) To be more speciﬁc let for < ε0 the number N denote the number of occurrences of in the Cantor normal form for . Let cα (n) := |{ < α : N = n}|. For small values of α the function cα can be evaluated by hands. For α = d already some calculation (see, for example, Theorem 2.48 in [8]) is needed to verify that c d (n) ∼

1 n d −1 . d !(d − 1)!

If one goes to α = the result is even more exciting, since we obtain √ 2n e 3 (1) c (n) ∼ √ . 4 3n (Here and in the sequel we denote by ∼ asymptotic equivalence. Thus an ∼ bn means as usual limn→∞ abnn = 1.) Now one can make a principal decision. Either one does not like such formulae at all or one get’s excited and wants to know how to obtain and how to apply them. (A simple proof of the formula (1) by induction seems ¨ managed to provide an elementary hopeless at least at ﬁrst. But still Erdos proof of (1) in [13]. We would also would like to mention that a Tauberian proof of (1) has been given in [19].) Mathematical experience from the last century (and even before) suggests that attacking diﬃcult counting problems is best done by studying related generating functions. Odlyzko commented for example in [24]:“Analytic methods are extremely powerful and when they apply, they often yield estimates of unparalleled precision.” Therefore let ∞ cα (n)z n . Cα (z) := n=0

240

ANDREAS WEIERMANN

Then under some standard conditions for r < 1 it holds that 1 Cα (z) (2) dz. cα (n) = 2i |z|=r z n+1 By comparing the coeﬃcients of the involved power series we see that (3)

C (z) =

∞ i=1

1 1 − zi

where C (z) is (uniformly compact) converging for |z| < 1. By exploiting (2) and (3) Hardy and Ramanujan (already in 1917) proved the formula (1) by introducing the circle method. In fact they proved a much stronger result which later was brought into perfection by Rademacher (see, for example, [3] for an accessible presentation) who provided a closed expression for c (n).

⎞ ⎛ 1 2 1 − sinh ∞ k 3 24 1 √ d ⎜ ⎟ kAk (n) ⎜ c (n) = √ ⎟ ⎝ ⎠ d 1 2 k=0 − 24 =n where Ak (n) =

h,k e −

2hn k

0 0. d +1 (1− )κd 1 . Since Let f(s) = (1 − )κ s d and h(v) = v d U (u) = exp (1 + o(1)) · u d +1 , d

we can choose a subset of S ⊆ {α < } such that cS (≤ v) ∼

exp(vh(v) + f(h(v))) h(v) 2f (h(v))

as v → ∞ (where f denotes the second derivative of f and cS (≤ v) the number of elements in S with norm not exceeding v. Note that the term on the right hand side is exactly of the form of equation (21.7) in [22].) Then Laplace’s method (cf. exercise 208 on page 80 in [30]) yields

∞ ∞ 1 −su −sn (5) e cS (≤ u)du = cS (n)e ∼ exp (1 − )κ d s· s 0 n=0

as s → 0 (where cS (≤ v) denotes the number of elements in S with norm equal to v). Let M (S) be the set of ﬁnite (normal form) sums (multisets) of elements d so that c d (n) ≥ from S. Then M (S) is contained in α : α < cM (S) (n). So it suﬃces to get a good lower bound on cM (S) (n). Let CS (s) :=

∞

cS (n)e −sn ,

n=0 ∞ ∞ 1 CS (sm) H (s) := m m=1

n=0

∞

and assume that H (s) = n=0 c (n)e −sn . Then coeﬃcientwise c (n) ≥ cS (n). Therefore the n-th coeﬃcient in exp(H (s)) is not smaller than the n-th coefﬁcient, say c (n), in exp(CS (s)). Hence CM (S) (n) ≥ c (n). Let U (u) := i≤u c (i). We now have by (5) ∞ s· e −su U (u)du =

∞ n=0

0

c (n)e −sn

1 = exp (1 + o(1))exp((1 − ))κ d . s

248

ANDREAS WEIERMANN

Then we obtain by the Tauberian theorem of Wagner [34] that u log(U (u)) ∼ (1 − )κ · 1 , (log(u)) d and hence the assertion, if a corresponding Tauberian condition is satisﬁed. This is indeed so since for g(s) = exp((1 + )κ s1d ) we have 0 < g(s) → ∞ and for some ﬁxed (g (s))2 ≤1 0 0. Let 1d d expk−1 ((1 − )κt) s dt f(s) = t d +1 1 and κ(1 − ) . h(v) = d logk−1 (v) (Note that again this term is exactly of the form of equation (21.7) in [22].) Since by induction hypothesis

u U (u) ≥ expk−1 (1 − o(1))κ logk−3 (u) we can choose a subset of T ⊇ {α < k−1 } such that cTS (≤ v) ∼

exp(vh(v) + f(h(v))) . h(v) 2f (h(v))

Then Laplace’s method yields

∞ ∞ 1 −su −sn (6) e cS (≤ u)du = cS (n)e ∼ expk−1 (1 − )κ d s· s 0 n=0

as s → 0.

ANALYTIC COMBINATORICS OF THE TRANSFINITE

249

Let M (S) be the set of ﬁnite (normal form) sums (multisets) of elements from S. Then M (S) is contained in {α : α < k (d )} so that ck (d ) (n) ≥ cM (S) (n). So it suﬃces to get a good lower bound on cM (S) (n). As before ∞ ∞ let CS (s) := n=0 cS (n)e −sn , let H (s) := m=0 m1 CS (sm) and assume that ∞ H (s) = n=0 c (n)e −sn . Then coeﬃcientwise c (n) ≥ cS (n). Therefore the n-th coeﬃcient in exp(H (s)) is not smaller than the n-th coeﬃcient, say c (n), in exp(CS (s)). Hence CM (S) (n) ≥ c (n). Let U (u) := i≤u c (i). We now have by (6) ∞ s· e −su cM (S) (≤ u)du =

∞ n=0

= exp

0

cM (S) (n)e −sn = exp(H (s)) ≥ exp(CS (s)) ∞

cS (n)e

−sn

=

n=0

∞

c (n)e −sn

n=0

1 . = exp((1 + o(1))) · expk−1 κ (1 − ) d s The Tauberian theorem of Wagner [34] yields log(U (u)) ∼ (1 − )κ · u 1 , hence the assertion, if a corresponding Tauberian condition is (log(k−2) (u)) d satisﬁed. This is indeed true since for g(s) = expk−1 (1 − )κ s1d we have 0 < g(s) → ∞ and for some ﬁxed 0 αn then let MAl α := 2MAl −1 α1 + · · · + 2MAl −1 αn . We guess that the asymptotic behavious of the resulting count function behaves similarly as the asymptotic of c Al .

ANALYTIC COMBINATORICS OF THE TRANSFINITE

259

Still another coding of ordinals less than l +1 for ﬁxed l has roots going back to Hardy and Ramanujan [18]. Deﬁnition 7. 1. If α = k1 + · · · + kn < with k1 ≥ · · · ≥ kn then HR1 α := p1k1 +1 · · · · · pnkn +1 . 2. For α = α1 + · · · + αn < l +1 (where l > 1) with α1 ≥ · · · ≥ αn put HR α HR α HRl α := p1 l −1 1 ·· · ··pn l −1 n if α = α1 +· · ·+ αn ≥ α1 ≥ · · · ≥ αn . For ≤ l +1 put

cHRl (n) := α < : HRl α ≤ n . log(n) d 1 1 Theorem 5. 1. cHR . d (n) ∼ d !d ! log(log(n)) log(n) 2 1 2. log(cHR . (n)) ∼ √ log(log(n)) 3 Proof. The ﬁrst assertion follows e.g. from Karamata’s theorem as shown in [36]. The ﬁrst assertion can also be proved by a more elementary argument of Schlage Puchta (private communication). The second assertion was proved by Hardy and Ramanujan [18].

We expect that Parameswaran’s theorem provides weak asympotics for the count functions related to higher -powers. Finally we would like to mention a (perhaps somewhat artiﬁcial) norm function which leads to fractional exponents for partitions. Deﬁnition 8. 1. Nk 0 := 0 2. Nk α := (1 + Nk α1 )k + · · · + (1 + Nk αn )k if α = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn . We guess that the asymptotic for the resulting count functions can be determined using Petrogradsky’s theorem starting at its base with d

cNdk (≤ n) ∼ constant · n k . The corresponding multiplicative analogue will be given by Deﬁnition 9. 1. Sk 0 := 1 2. Sk α := pSkk α1 · · · · · pSkk αn if α = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn . We expect that the asymptotic will start with something like d

cSkd (≤ n) ∼ constant · (log n) k . The choice of Nk and Sk is of course somewhat artiﬁcial. But, and this seems to be of certain interest, if one uses Nk in place of N and Sk in place of S in Theorems 7 and 8 of the next section it will turn out that the threshold function will not change. So there will be a phenomenon of structural stability concerning applications of diﬀerent norms.

260

ANDREAS WEIERMANN

§4. Applications to phase transition results for G¨odel incompleteness. We start with some standard conventions. As usual the least integer greater than or equal to a given real number x is denoted by x and similarly the greatest integer smaller than or equal to a given real number x is denoted by x. The binary length |n| of a natural number n is deﬁned by |n| := log(n + 1). The d -times iterated length function |·|d is deﬁned recursively as follows |x|0 := x and |x|d +1 := ||x|d |. (This function mainly serves as an LPA deﬁnable version of log(d ) .) As usual we assume that the ordinals less than ε0 are available in PA via a standard coding. Recall that for α < ε0 the number Nα denotes the number of occurrences of in the Cantor normal form of α. Let CWO(N, α, f) be the following principle ∀K ∃M B(N, α, f, K, M ) where B(N, α, f, K, M ) is the assertion ! ! " (∀α0 , . . . , αM < α) (∀i ≤ M ) Nαi ≤ K + f(i) =⇒ ! "" (∃i, j < ) 0 ≤ i < j ≤ M & αi ≤ αj . Theorem 6 (Friedman). Let d ≥ 0 be a ﬁxed natural number. Then IΣd +1 CWO(N, d +2 , f) where f is the identity map. Proof. See, for example, [33] for a proof.

Before we can sharpen this result we need to recall the deﬁnition of the standard system of canonical fundamental sequences and the Hardy hierarchy. Deﬁnition 10. 1. If = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn and αn = αn + 1 then [x] := α1 + · · · + αn · x. 2. If = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn and αn is a limit then [x] := α1 + · · · + αn [x] . Deﬁnition 11 (The Hardy hierarchy). H0 (x) := x,

Hα+1 (x) := Hα (x + 1),

H (x) := H [x] (x) if is a limit.

The basic theory of the Hardy hierarchy is developed, for example, in [7] or [31]. The most fundamental result concerning this hierarchy is that (Hα )α α1 > · · · > αM with Nαi ≤ K + i and α0 < d +2 . Our aim is to transform this sequence into a strictly descending sequence −1 0 > 1 > · · · > M with N i ≤ p(K ) + |i| · Hd +2 (i) |i|d for i ≤ M where p is a primitive recursive function depending only on K . If this is achieved we are done since M depends on K in a way which is not provably recursive in IΣd +1 . From Nα0 ≤ K we conclude α0 < d +1 (K − d − 1). For the sequel assume that D depending primitive recursively on K is large enough so that the asymptotic bounds apply. For i ≥ D put Mi := α < d +1 (K + 3) : Nα ≤ |i| · K+1 |i|d and let enumi (l ) be the l -th element of Mi . Then we deﬁne i := d +1 (K + 3) · α|i| + enumi 2|i| − i .

262

ANDREAS WEIERMANN

(This transformation might be interpreted as a renormalization operator on descending sequences.) The sequence i is strictly decreasing. Moreover we ﬁnd for i ≥ D Ni ≤ (d + 1 + K + 3) · (K + |i|) + |i| · K+1 |i|d ≤ K + |i| · K |i|d . Now we sharpen this slightly using a trick due to Arai [4]. For i ≤ F (K ) we (i) ≤ F −1 (i) ≤ F −1 (F (K )) = K . Therefore have H−1 d +2 −1 Ni ≤ K + |i| · Hd +2 (i) |i|d holds for i ≥ D. Now we have to check the side conditions. For i < D we simply put i := d +1 (K + 3) · d +1 (K + 3) + D − i. We are nearly through but one last point has to be checked namely the well-deﬁnedness of i . For this it suﬃces to check that Mi has at least 2|i| many elements. This is the place where we need the bounds from analytic combinatorics. Case 1. d = 0. We have by assertion 1 of Corollary 1 and obvious asymptotic calculations for a suitable and constant C0,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable constant D0,K K+2 |Mi | ≥ C0,K · |i| · K+1 |i|d √ K+2 K+1 ≥ C0,K · i K+2

≥ C0,K · i K+1 ≥ 2|i| . Case 2. d = 1. We have by assertion 2 of Corollary 1 and obvious asymptotic calculations for a suitable and a constant C1,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable constant D1,K √ K+3 K+4 C1,K |i|·

|Mi | ≥ 2

K+3

K+4 C1,K |i|

≥2

K+3 K+4

≥ 2C1,K ≥ 2|i| .

K+1

1+

|i|

1 K+1

K+2 · K+3 K+4

|i| K+1

K+3 K+4

ANALYTIC COMBINATORICS OF THE TRANSFINITE

263

Case 3. d ≥ 2. We have by assertions 3 and 4 of Corollary 1 and obvious asymptotic calculations for a suitable and a constant Cd,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable (with respect to K ) constant Dd,K √ |i|· K+1 |i|d √ Cd,K √ K+3 ||i|· K+1 |i| | d d −1 |Mi | ≥ 2 ≥ 2|i| since K+1 |i|d ||i| · K+1 |i|d |d −1 → 0 as i → ∞. Now we prove the second assertion. Fix < d +2 . Given K let M := 2H (K·2)+1 . This is well deﬁned in IΣd +1 . Moreover we may assume that M is large enough for asymptotic purposes. We claim that this M is suﬃcient for our purposes. Assume otherwise that α0 > α1 > · · · > αM is a sequence such −1 that Nαi ≤ K + |i| · H (i) |i|d and α0 < d +2 . Then α0 < d +1 (K − d − 1). We have for i ≥ M2 that H −1 (i) ≥ H −1 (H (2K )) = 2K and hence for i ≥ n2 αi ∈ α < d +1 (K ) : Nα ≤ |M | 2K |M |d . # K+3

The number of elements of the latter set is at least M2 as witnessed by the αi . We obtain the contradiction by calculating bounds on the number of elements on this set. To this end we distinguish three cases. Case 1. d = 0. By choice of M we may assume that the bounds of assertion 1 of Corollary 1 apply and that for a certain constant C0,K we have √ K M 2K < C0,K · |M | M 2 √ = C0,K · (|M |)K M M . < 2 Case 2. d = 1. By choice of M we may assume that the bounds from assertion 2 of Corollary 1 apply and that for a certain constant C1,K we have K √ K+1 2K M < 2C1,K · |M | |M | 2 = 2C1,K ·(|M |) M . < 2

2K+1 K 2K K+1

264

ANDREAS WEIERMANN

Case 3. d ≥ 2. By choice of M we may assume that the bounds from assertions 3 and 4 of Corollary 1 apply and that for a certain constant Cd,K we have √ Cd,K · |M | 2K |M |d √ K M |M | 2K |M |d −1 d −1

March 23, 2010

11:41

978 0 521 76065 2

This page intentionally left blank

ii

Logic Colloquium 2007 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 2007, 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. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay’s conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

Fran¸coise Delon was Directrice d’´etudes at the Centre de Formation des PEGC of Reims and Humboldt Stipendiatin at Freiburg and is presently a Directrice de Recherche at Centre National de la Recherche Scientifique. Ulrich Kohlenbach is a Professor of Mathematics at TU Darmstadt (Germany). He is the coordinating editor of Annals of Pure and Applied Logic and the president of the Deutsche Vereinigung f¨ur Mathematische Logik und f¨ur Grundlagen der Exakten Wissenschaften. Penelope Maddy is a Distinguished Professor of Logic and Philosophy of Science at the University of California, Irvine. She is a Fellow of the American Academy of Arts and Sciences and is currently the president of the Association for Symbolic Logic. Frank Stephan is an Associate Professor in the departments of mathematics and computer science at the National University of Singapore. He is the editor of the Journal of Symbolic Logic.

CUUS1074-Lerman

March 23, 2010

11:41

978 0 521 76065 2

ii

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 dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board H. Dugald Macpherson, Managing Editor Department of Pure Mathematics, School of Mathematics, University of Leeds Jeremy Avigad Department of Philosophy, Carnegie Mellon University Vladimir Kanovei Institute for Information Transmission Problems, Moscow Manuel Lerman Department of Mathematics, University of Connecticut Heinrich Wansing Institute of Philosophy, Dresden University of Technology Thomas Wilke Institut f¨ur Informatik, Christian-Albrects-Universitat zu Kiel

More information, including a list of the books in the series, can be found at http://www.aslonline.org/books-lnl.html.

Logic Colloquium 2007 Edited by

FRANC¸ OISE DELON UFR de Math´ematiques

ULRICH KOHLENBACH Technische Universit¨at Darmstadt

PENELOPE MADDY University of California, Irvine

FRANK STEPHAN National University of Singapore

association for symbolic logic

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/9780521760652 © Association of Symbolic Logic 2010 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 2010 ISBN-13

978-0-511-78785-0

eBook (EBL)

ISBN-13

978-0-521-76065-2

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.

CONTENTS Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Speakers and Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

ˇ ci´c, Pavel Pudl´ak, Greg Restall, Alasdair Urquhart, and Vedran Caˇ Albert Visser Decorated linear order types and the theory of concatenation . . . . . . .

1

Andr´es Eduardo Caicedo Cardinal preserving elementary embeddings . . . . . . . . . . . . . . . . . . . . . . . .

14

Fernando Ferreira Proof interpretations and majorizability . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Philipp Gerhardy Proof mining in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Steve Jackson Cardinal structure under AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

Bakhadyr Khoussainov and Mia Minnes Three lectures on automatic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Ya’acov Peterzil Pillay’s conjecture and its solution—a survey . . . . . . . . . . . . . . . . . . . . . . . 177 Greg Restall Proof theory and meaning: On the context of deducibility . . . . . . . . . . 204 Marcus Tressl Bounded super real closed rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Andreas Weiermann Analytic combinatorics of the transﬁnite: A unifying Tauberian perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

vii

INTRODUCTION

The Logic Colloquium 2007, the European Summer Meeting of the Association for Symbolic Logic, was held in Wrocław, Poland, from 14 to 19 July 2007. It was colocated with the following events: the thirty-fourth International Colloquium on Automata, Languages and Programming (ICALP); the twenty-second Annual IEEE Symposium on Logic in Computer Science (LICS) and the ninth ACM-SIGPLAN International Symposium on Principles and Practice of Declarative Programming (PPDP). There was an agreement with LICS on running joint sessions for one day. More than 200 participants from all over the world took part in the Logic Colloquium. The programme consisted of 3 tutorials, 11 invited plenary talks, 6 joint talks with LICS (2 long, 4 short) and 21 talks in 5 special sessions on set theory, proof complexity and nonclassical logics, philosophical and applied logic at the JPL, logic and analysis and model theory. In addition to these invited talks, there were 63 contributed talks. The programme committee consisted of Alessandro Andretta (Turin), Franc¸oise Delon (Paris 7), Ulrich Kohlenbach (Darmstadt), Steﬀen Lempp (Madison, Chair), Penelope Maddy (UC Irvine), Jerzy Marcinkowski (Wrocław), Ludomir Newelski (Warsaw), Andrew Pitts (Cambridge), Pavel Pudl´ak (Prague), Sławomir Solecki (Urbana-Champaign), Frank Stephan ¨ (Singapore) and Goran Sundholm (Leiden). The local organizing committee consisted of Tobias Kaiser, Piotr Kowalski, Jan Kraszewski, Amador MartinPizarro, Serge Randriambololona and Roman Wencel. The Logic Colloquium 2007 wants to acknowledge its sponsors for their generous support of the event: the Association for Symbolic Logic, the Polish Academy of Sciences and the University of Wrocław. The next pages give an overview of the programme of the meeting; in addition to the talks listed, there were contributed talks from all ﬁelds of mathematical logic. Speakers invited to give a plenary or special session talk were also ix

x

INTRODUCTION

invited to contribute to this volume. The contributions handed in were refereed according to the high standards of logic journals. We would like to thank the authors for their excellent contributions as well as the referees for their diligent work to review and evaluate the submissions. The Editors Franc¸oise Delon Ulrich Kohlenbach Penelope Maddy Frank Stephan

SPEAKERS AND TITLES

Tutorials Steve Jackson (North Texas): Cardinal arithmetic in L(R). Bakh Khoussainov (Auckland): Automatic structures. Kobi Peterzil (Haifa): The inﬁnitesimal subgroup of a deﬁnably compact group.

Invited Plenary Talks Albert Atserias (Barcelona): Structured ﬁnite model theory. Matthias Baaz (Vienna): Towards a proof theory of analogical reasoning. Vasco Brattka (Cape Town): Computable analysis and eﬀective descriptive set theory. Zo´e Chatzidakis (Paris 7): Model theory of diﬀerence ﬁelds and some applications. Gabriel Debs (Paris 6): Coding compact spaces of Borel functions. Fernando Ferreira (Lisbon): On a new functional interpretation. Andrzej Grzegorczyk (Warsaw): Philosophical content of formal achievements. Bjørn Kjos-Hanssen (Hawaii): Brownian motion and Kolmogorov complexity. Piotr Kowalski (Wrocław): Deﬁnability in diﬀerential ﬁelds. Paul Larson (Miami U): Large cardinals and forcing-absoluteness. Tony Martin (UC Los Angeles): Sets and the concept of set. xi

xii

SPEAKERS AND TITLES

LICS Joint Long Talks Martin Hyland (Cambridge): Combinatorics and proofs. Colin Stirling (Edinburgh): Higher-order matching, games and automata.

LICS Joint Short Talks Cristiano Calcagno (Imperial College): Can logic tame systems programs? Mart´ın Escardo´ (Birmingham): Inﬁnite sets that admit exhaustive search. Rosalie Iemhoﬀ (Utrecht): Skolemization in constructive theories. Alex Simpson (Edinburgh): Non-well-founded proofs.

Special Session Set Theory Organized by Ilijas Farah and Joel Hamkins. M´arton Elekes (Hungarian Academy of Sciences): Partitioning κ-fold covers into κ many subcovers. ¨ Gunter Fuchs (Munster): Maximality principles for closed forcings. Victoria Gitman (City University of New York): Scott’s problem for proper Scott sets. Lionel Nguyen Van The (Calgary): The Urysohn sphere is oscillation stable. Asger Tornquist (Toronto): Classifying measure preserving actions up to conjugacy and orbit equivalence. Matteo Viale (Vienna): The constructible universe for the anti-foundation axiom system ZFA.

Special Session Proof Complexity and Nonclassical Logics Organized by Pavel Pudl´ak. Emil Jer´abek (Prague): Proof systems for modal logics. George Metcalfe (Vanderbilt): Substructural fuzzy logics. Alasdair Urquhart (Toronto): Complexity problems for substructural logics.

SPEAKERS AND TITLES

xiii

Special Session Philosophical and Applied Logic at the JPL Organized by Penelope Maddy. Aldo Antonelli (UC Davis): From philosophical logic to computer science. Horacio Arlo` Costa (Carnegie-Mellon): Philosophical logic meets formal epistemology. Greg Restall (Melbourne): Proof theory and meaning: three case studies. Albert Visser (Utrecht): Interpretations in philosophical logic.

Special Session Logic and Analysis Organized by Ita¨ı Ben Yaacov and Ulrich Kohlenbach. Philipp Gerhardy (Pittsburgh/Oslo): Local stability of ergodic averages. ¨ Peter Hertling (Univ. der Bundeswehr Munchen): Computability and non-computability results for the topological entropy of shift spaces. Julien Melleray (Urbana-Champaign): Geometry of the Urysohn space: a model-theoretic approach. Andreas Weiermann (Ghent): Analytic combinatorics of the transﬁnite.

Special Session Model Theory Organized by Franc¸oise Delon and Ludomir Newelski. Vera Djordjevic (Uppsala): Independence in structures and ﬁnite satisﬁability. Amador Martin-Pizarro (Lyon 1): Some thoughts on bad objects. Ziv Shami (Tel Aviv): Countable imaginary simple unidimensional theories. Marcus Tressl (Regensburg): Super real closed rings.

DECORATED LINEAR ORDER TYPES AND THE THEORY OF CONCATENATION

ˇ CI ˇ C, ´ PAVEL PUDLAK, ´ VEDRAN CA GREG RESTALL, ALASDAIR URQUHART, AND ALBERT VISSER

Abstract. We study the interpretation of Grzegorczyk’s Theory of Concatenation TC in structures of decorated linear order types satisfying Grzegorczyk’s axioms. We show that TC is incomplete for this interpretation. What is more, the ﬁrst order theory validated by this interpretation interprets arithmetical truth. We also show that every extension of TC has a model that is not isomorphic to a structure of decorated order types. We provide a positive result, to wit a construction that builds structures of decorated order types from models of a suitable concatenation theory. This construction has the property that if there is a representation of a certain kind, then the construction provides a representation of that kind.

§1. Introduction. In his paper [2], Andrzej Grzegorczyk introduces a theory of concatenation TC. The theory has a binary function symbol ∗ for concatenation and two constants a and b. The theory is axiomatized as follows. TC1. (x ∗ y) ∗ z = x ∗ (y ∗ z) TC2. x ∗ y = u ∗ v → ((x = u ∧ y = v) ∨ ∃w ((x∗w = u∧y = w∗v)∨(x = u∗w∧y∗w = v))) TC3. x ∗ y = a TC4. x ∗ y = b TC5. a = b Axioms TC1 and TC2 are due to Tarski [7]. Grzegorczyk calls axiom TC2 the editor axiom. We will consider two weaker theories. The theory TC0 has the signature with just concatenation, and is axiomatized by TC1,2. The theory TC1 is axiomatized by TC1,2,3. We will also use TC2 for TC. ´ ¸ z` a of the Some of the results of this note were obtained during the Excursion to mountain Sle inspiring Logic Colloquium 2007 in Wrocław and, in part, in the evening after the Excursion. We thank the organizers for providing this wonderful opportunity. We thank Dana Scott for his comments, insights and questions. We are grateful to Vincent van Oostrom for some perceptive remarks. Pavel Pudl´ak was supported by grants A1019401 and 1M002162080. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

1

2

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

The theories we are considering have various interesting interpretations. First they are, of course, theories of strings with concatenation; in other words, theories of free semigroups. Secondly they are theories of wider classes of structures, to wit structures of decorated linear order types, which will be deﬁned below1 . The theories TCi are theories for concatenation without the empty string, i.e., without the unit element. Adding a unit ε one obtains another class of theories TCεi , theories of free monoids, or theories of structures of decorated linear order types including the empty linear decorated order type. The basic list of axioms is as follows. TCε 1. ε ∗ x = x ∧ x ∗ ε = x TCε 2. (x ∗ y) ∗ z = x ∗ (y ∗ z) TCε 3. x ∗y = u ∗v → ∃w ((x ∗w = u ∧y = w ∗v)∨(x = u ∗w ∧w ∗y = v)) TCε 4. a = ε TCε 5. x ∗ y = a → (x = ε ∨ y = ε) TCε 6. b = ε TCε 7. x ∗ y = b → (x = ε ∨ y = ε) TCε 8. a = b. We take TCε0 to be the theory axiomatized by TCε 1, 2, 3, TCε1 to be TCε0 + TCε 4, 5 and TCε := TCε2 to be TCε1 + TCε 6, 7, 8. One can show that TC is bi-interpretable with TCε , in which a unit ε is added via one dimensional interpretations without parameters. The theory TC1 is biinterpretable with TCε1 via two-dimensional interpretations with parameters. The situation for TC0 seems to be more subtle. See also [10]. In Section 6, we will study an extension of TCε0 . Andrzej Grzegorczyk and Konrad Zdanowski have shown that TC is essentially undecidable. This result can be strengthened by showing that Robinson’s Arithmetic Q is mutually interpretable with TC. Note that TC0 is undecidable —since it has an extension that parametrically interprets TC— but that TC0 is not essentially undecidable: it is satisﬁed by a one-point model. Similarly TC1 is undecidable, but it has as an extension the theory of ﬁnite strings of a’s, which is a notational variant of Presburger Arithmetic and, hence, decidable. We will call models of TC0 concatenation structures, and we will call models of TCi concatenation i-structures. The relation of isomorphism between concatenation structures will be denoted by ∼ =. We will be interested in concatenation structures, whose elements are decorated linear order types with the operation concatenation of decorated order types. Let a non-empty class A be given. An A-decorated linear ordering is a structure D, ≤, f , where D is a non-empty domain, ≤ is a linear ordering on D, and f is a function from 1 A special case of decorated linear order types is addition of sets as discovered by Tarski (see [8]). It is shown by Laurence Kirby in [3] that the structure of addition on sets is isomorphic to addition of well-founded order types with a proper class of decorating objects.

DECORATED LINEAR ORDER TYPES

3

D to A. A mapping φ is an isomorphism between A-decorated linear order types D, ≤, f and D , ≤ , f iﬀ it is a bijection between D and D such that, for all d, e in D, d ≤ e ⇔ φd ≤ φe, and fd = f φd . Our notion of isomorphism gives us a notion of A-decorated linear order type. We have an obvious notion of concatenation between A-decorated linear orderings which induces a corresponding notion of concatenation for A-decorated linear order types. We use α, , . . . to range over such linear order types. Since, linear order types are classes we have to follow one of two strategies: either to employ Scott’s trick to associate a set object to any decorated linear order type or to simply refrain from dividing out isomorphism but to think about decorated linear orderings modulo isomorphism. We will employ the second strategy. We will call a concatenation structure whose domain consists of (representatives of) A-decorated order types, for some A, and whose concatenation is concatenation of decorated order types: a concrete concatenation structure. It seems entirely reasonable to stipulate that e.g. the interpretation of a in a concrete concatenation structure is a decorated linear order type of a one element order. However, for the sake of generality we will refrain from making this stipulation. Grzegorczyk conjectured that every concatenation 2-structure is isomorphic to a concrete concatenation structure. We prove that this conjecture is false. (i) Every extension of TC1 has a model that is not isomorphic to a concrete concatenation 1-structure and (ii) the set of principles valid in all concrete concatenation 2-structures interprets arithmetical truth. The plan of the paper is as follows. We show, in Section 2, that we have, for all decorated order types α, and , the following principle: (†)

∗ α ∗ = α ⇒ ∗ α = α ∗ = α.

This fact was already known. It is due to Lindenbaum, credited to him in ´ Sierpinski’s book [6] on p. 248. It is also problem 6.13 of [4]. It is easy to see that every group is a concatenation structure and that (†) does not hold in the two element group. We show, in Section 5, that every concatenation structure can be extended to a concatenation structure with any number of atoms. It follows that there is a concatenation structure with at least two atoms in which (†) fails. Hence, TC is incomplete for concrete concatenation structures. In Section 3, we provide a counterargument of a diﬀerent ﬂavour. We provide a tally interpretation that deﬁnes the natural numbers (with concatenation in the role of addition) in every concrete concatenation 2structure. It follows that every extension of TC1 is satisﬁed by a concatenation 1-structure that is not isomorphic to any concrete concatenation 1-structure, to wit any model of that extension that contains a non-standard element. In Section 4, we strengthen the result of Section 3, by showing that in concrete concatenation 2-structures we can add multiplication to the natural numbers.

4

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

It follows that the set of arithmetically true sentences is interpretable in the concretely valid consequences of TC2 . Finally, in Section 6 we prove a positive result. We provide a mapping from arbitrary models of a variant of an extension of TC0 to structures of decorated order types. As we have shown such a construction cannot always provide a representation. We show that, for a restricted class of representations, we do have: if a model has a representation in the class, then the construction yields such a representation. §2. A principle for decorated order types. In this section we prove a universal principle that holds in all concatenation structures, which is not provable in TC. There is an earlier proof of this principle [4, p. 187]. Our proof, however, is diﬀerent from that of Komj´ath and Totik. Theorem 1. Let α0 , α1 , α2 be decorated order types. Suppose that α1 = α0 ∗ α1 ∗ α2 . Then, α1 = α0 ∗ α1 = α1 ∗ α2 . Proof. Suppose α1 = α0 ∗ α1 ∗ α2 . Consider a decorated linear ordering A := A, ≤, f of type α1 , By our assumption, we may partition A into A0 , A1 , A2 , such that:

A, ≤, f = A0 , ≤ A0 , f A0 ∗ A1 , ≤ A1 , f A1 ∗ A2 , ≤ A2 , f A2 , where Ai := Ai , ≤ Ai , f Ai is an instance of αi , Let φ : A → A1 be an isomorphism. Let φ n A(i) := φ n [A(i) ], ≤ φ n [A(i) ], f φ n [A(i) ] . We have: φ n Ai is of order type αi and φ n A is of order type α1 . Clearly, φA0 is an initial substructure of φA = A1 . So, A0 and φA0 are to the right of A0 . Similarly, for φ n A0 and φ n+1 A0 . disjoint and φA 0 adjacent i Take A0 := i∈ φ A0 . We ﬁnd that A 0 := A0 , ≤ A0 , f A0 is initial in A and of decorated linear order type α0 . So α1 = α0 ∗ , for some . It follows that α0 ∗ α1 = α0 ∗ α0 ∗ = α0 ∗ = α1 . The other identity is similar. So, every concrete concatenation structure validates that α1 = α0 ∗ α1 ∗ α2 implies α1 = α0 ∗ α1 = α1 ∗ α2 . We postpone the proof that this principle is not provable in TC to Section 5. §3. Deﬁnability of the natural numbers. In this section, we show that the natural numbers can be deﬁned in every concrete concatenation 1-structure. We deﬁne: • • • •

x ⊆ y :↔ x = y ∨ ∃u (u ∗ x = y)∨ ∃v (x ∗ v = y)∨ ∃u, v (u ∗ x ∗ v = y). x ⊆ini y :↔ x = y ∨ ∃v (x ∗ v = y). x ⊆end y :↔ x = y ∨ ∃u (u ∗ x = y). a ) :↔ ∀m⊆ini n (m = a ∨ ∃k (k = m ∧ m = k ∗ a)). (n : N

DECORATED LINEAR ORDER TYPES

5

a is derived from the analogous use in type theory. We The use of ‘:’ in n : N a for: (m : N a ) ∧ (n : N a ), could read it as: n is of sort Na . We write m, n : N a a with the extension of N etc. In the context of a structure we will confuse N in that structure. We prove the main theorem of this section. Theorem 2. In any concrete concatenation 1-structure, we have: a = an+1 | n ∈ . N a is precisely the class of natural numbers in tally representation In other words, N (starting with 1). Note that ∗ on this set is addition. Proof. Consider any concrete concatenation 1-structure A. It is easy to a . Clearly, every element x of N a is either a or it see that every an+1 is in N has a predecessor, i.e., there is a y such that x = y ∗ a. The axioms of TC1 guarantee that this predecessor is unique. This justiﬁes the introduction of the a . Let α be the order type corresponding partial predecessor function pd on N to a. Let 0 be any element of Na . If, for some n, pdn 0 is undeﬁned, then 0 is clearly of the form α k+1 , for k in . We show that the other possibility cannot obtain. Suppose n := pdn 0 is always deﬁned. Let A be a decorated linear ordering of type α and let Bi be a decorated linear ordering of type i . We assume that the domain A of A is disjoint from the domains Bi of the Bi . Thus, we may implement Bi+1 ∗ A just by taking the union of the domains. Let φi be isomorphisms from Bi+1 ∗ A to Bi . Let Ai := (φ0 ◦ · · · ◦ φi )(A). Then, the Ai are all of type α and, for some C, we have B0 ∼ = C ∗ · · · ∗ A1 ∗ A 0 . Similarly B1 ∼ = C ∗ · · · ∗ A2 ∗ A1 . Let ˘ be the opposite ordering of . a2. A It follows that 0 = ∗ α ˘ = 1 = pd(0 ). Hence, 0 is not in N contradiction. We call a concatenation structure standard if Na deﬁnes the tally natural numbers. Since, by the usual argument, any extension of TC1 has a model with non-standard numbers, we have the following corollary. Corollary 3. Every extension of TC1 has a model that is not isomorphic to a concrete concatenation 1-structure. In a diﬀerent formulation: for every concatenation 1-structure there is an elementarily equivalent concatenation 1structure that is not isomorphic to a concrete concatenation 1-structure. Note that the non-negative tally numbers with addition form a concrete concatenation 1-structure. Thus, the concretely valid consequences of TC1 + a ), i.e., the principles valid in every concrete concatenation 1∀x (x : N a ) are decidable. structure satisfying ∀x (x : N 2 Note

that we are not assuming that is in A.

6

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

§4. Deﬁnability of multiplication. If we have two atoms to work with, we can add multiplication to our tally numbers. This makes the set of concretely valid consequences of TC non-arithmetical. The main ingredient of the deﬁnition of multiplication is the theory of relations on tally numbers. In TC, we can develop such a theory. The development has some resemblance to the construction in the classic paper of Quine [5]. However, the ideas here are somewhat more intricate, since we areworking in a more generalcontext than that of [5]. We represent the relation x0 , y0 , . . . , xn−1 , yn−1 , by: bb ∗ x0 ∗ b ∗ y0 ∗ bb ∗ x1 ∗ . . . bb ∗ xn−1 ∗ b ∗ yn−1 ∗ bb. We deﬁne: • r : REL :↔ bb ⊆end r, • ∅ := bb, a ∧ bb ∗ x ∗ b ∗ y ∗ bb ⊆ r. • x[r]y :↔ x, y : N • adj(r, x, y) := r ∗ x ∗ b ∗ y ∗ bb. Clearly, we have: TC ∀u, v ¬ u[∅]v. To verify that this coding works we need the adjunction principle. Theorem 4. We have: a) → TC (r : REL ∧ x, y, u, v : N u[adj(r, x, y)]v ↔ u[r]v ∨ (u = x ∧ v = y) . We can prove this result by laborious and unperspicuous case splitting. However, it is more elegant to do the job with the help of a lemma. Consider any model of TC0 . Fix an element w. We call a sequence (w0 , . . . , wk ) a partition of w if we have that w0 ∗ · · · ∗ wk = w. The partitions of w form a category with the following morphisms. f : (u0 , . . . , un ) → (w0 , . . . , wk ) iﬀ f is a surjective and weakly monotonic function from n + 1 to k + 1, such that, for any i ≤ k, wi = us ∗ · · · ∗ u , where f(j) = i iﬀ s ≤ j ≤ . We write (u0 , . . . , un ) ≤ (w0 , . . . , wk ) for: ∃f f : (u0 , . . . , un ) → (w0 , . . . , wk ). In this case we say that (u0 , . . . , un ) is a reﬁnement of (w0 , . . . , wk ). Lemma 5. Consider any concatenation structure. Let w be an element of the structure. Then, any two partitions of w have a common reﬁnement. Proof. Fix any concatenation structure. We ﬁrst prove that, for all w, all pairs of partitions (u0 , . . . , un ) and (w0 , . . . , wk ) of w have a common reﬁnement, by induction of n + k. If either n or k is 0, this is trivial. Suppose that (u0 , . . . , un+1 ) and (w0 , . . . , wk+1 ) are partitions of w. By the editor axiom, we have either (a) u0 ∗ · · · ∗ un = w0 ∗ · · · ∗ wk and un+1 = wk+1 , or there is a v such that (b) u0 ∗ · · · ∗ un ∗ v = w0 ∗ · · · ∗ wk and un+1 = v ∗ wk+1 , or (c) u0 ∗ · · · ∗ un = w0 ∗ · · · ∗ wk ∗ v and v ∗ un+1 = wk+1 . We only treat case (b), the other cases being easier or similar. By the induction hypothesis, there is a common reﬁnement (x0 , . . . xm ) of (u0 , . . . , un , v) and (w0 , . . . , wn ). Let this

DECORATED LINEAR ORDER TYPES

7

be witnessed by f, resp. g. It is easily seen that (x0 , . . . xm , wk+1 ) is the desired reﬁnement with witnessing functions f and g , where f := f[m+1 → n+1], g := g[m + 1 → k + 1]. Here f[m + 1 → n + 1] in the result of extending f to assign n + 1 to m + 1. We turn to the proof of Theorem 4. The veriﬁcation proceeds more or less as one would do it for ﬁnite strings. Proof. Consider any concatenation 2-structure. Suppose REL(r). The right-to-left direction is easy, so we treat left-to-right. Suppose x, y, u and v are tally numbers. and u[adj(r, x, y)]v. There are two possibilities. Either r = bb or r = r0 ∗ bb. We will treat the second case. Let s := adj(r, x, y). One the following four partitions is a partition of s: (i) (b, b, u, b, v, b, b), or (ii) (w, b, b, u, b, v, b, b), or (iii) (b, b, u, b, v, b, b, z), or (iv) (w, b, b, u, b, v, b, b, z). We will treat cases (ii) and (iv). Suppose := (w, b, b, u, b, v, b, b) is a partition of s. We also have that := (r0 , b, b, x, b, y, b, b) is a partition of s. Let (t0 , . . . , tk ) be a common reﬁnement of and , with witnessing functions f and g. The displayed b’s in these partitions must have unique places among the ti . We deﬁne m to be the unique i such that f(i) = m, provided that m = b. Similarly, for m . (To make this unambiguous, we assume that if = , we take as the common reﬁnement with f and g both the identity function.) We evidently have 7 = 7 = k and 6 = 6 = k − 1. Suppose 4 < 4 . It follows that b ⊆ v. So, v would have an initial subsequence that ends in b, which is impossible. So, 4 < 4 . Similarly, 4 < 4 . So 4 = 4 . It follows that v = y. Reasoning as in the case of 4 and 4 , we can show that 2 = 2 and, hence u = x. Suppose := (w, b, b, u, b, v, b, b, z) is a partition of s. We also have that := (r0 , b, b, x, b, y, b, b) is a partition of s. Let (t0 , . . . , tk ) be a common reﬁnement of and , with witnessing functions f and g. We consider all cases, where 1 < 6 . Suppose 6 = 1 + 1 = 2 . Note that 7 = 6 + 1, so we a . Suppose 2 < 6 < 4 . In this case we ﬁnd: b ⊆ x, quod non, since x is in N have a b as substring of x. Quod non. Suppose 6 = 4 . Since 7 = 6 + 1, we get a b in y. Quod non. Suppose 4 < 6 < 6 . In this case, we get a b in y. Quod impossible. Suppose 6 ≥ 6 = k − 1. In this place there is no place left for z among the ti . So, in all cases, we obtain a contradiction. So the only possibility is 6 ≤ 1 . Thus, it follows that u[r]v. We can now use our relations to deﬁne multiplication of tally numbers in the usual way. See e.g. Section 2.2 of [1]. In any concrete concatenation 2-structure, we can use induction to verify the deﬁning properties of multiplication as deﬁned. It follows that we can interpret all arithmetical truths in the set of concretely valid consequences of TC. Corollary 6. We can interpret true arithmetic in the set of all principles valid in concrete concatenation 2-structures.

8

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

§5. The sum of concatenation structures. In this section we show that concatenation structures are closed under sums. This result will make it possible to verify the claim that the universal principle of Section 2 is not provable in TC. The result has some independent interest, since it provides a good closure property of concatenation structures. Consider two concatenation structures A0 and A1 . We write for concatenation in the Ai . We may assume, without loss of generality, that the domains of A0 and A1 are disjoint. We deﬁne the sum B := A0 ⊕ A1 as follows. • The domain of B consists of non-empty sequences w0 · · · wn−1 , where the wj are alternating between elements of the domains of A0 and A1 . In other words, if wj is in the domain of Ai , then wj+1 , if it exists, is in the domain of A1−i . • The concatenation ∗ of := w0 · · · wn−1 and := v0 · · · vk−1 is w0 · · · wn−1 v0 · · · wk−1 , in case wn−1 and v0 are in the domains of diﬀerent structures Ai . The concatenation ∗ is w0 · · · (wn−1 v0 ) · · · wk−1 , in case wn−1 and v0 are in in the same domain. In case ∗ is obtained via the ﬁrst case, we say that and are glued together. If the second case obtains, we say that and are clicked together. Theorem 7. The structure B = A0 ⊕ A1 is a concatenation structure. Proof. Associativity is easy. We check the editor property TC2. Suppose

0 ∗ 1 = z0 · · · zm−1 = 0 ∗ 1 . We distinguish a number of cases. Case 1. Suppose both of the pairs 0 , 1 and 0 , 1 are glued together. Then, for some k, n > 0, we have 0 = z0 · · · zk−1 , 1 = zk · · · zm−1 , 0 = z0 · · · zn−1 , and 1 = zn · · · zm−1 . So, if k = n, we have 0 = 0 and 1 = 1 . If k < n, we have 0 = 0 ∗ (zk · · · zn−1 ) and 1 = (zk · · · zn−1 ) ∗ 1 . The case that n < k is similar. Case 2. Suppose 0 , 1 is glued together and that 0 , 1 is clicked together. So, there are k, n > 0, u0 , and u1 such that 0 = z0 · · · zk−2 u0 , 1 = u1 zk · · · zm−1 , u0 u1 = zk−1 , 0 = z0 · · · zn−1 , and 1 = zn · · · zm−1 . Suppose k ≤ n. Then, 0 = 0 ∗(u1 zk · · · zn−1 ) and 1 = (u1 zk · · · zn−1 )∗1 . Note that, in case k = n, the sequence zk · · · zn−1 is empty. The case that k ≥ n is similar. Case 3. This case, where 0 , 1 is clicked together and 0 , 1 is glued together, is similar to Case 2. Case 4. Suppose that 0 , 1 and 0 , 1 are both clicked together. So, there are k, n > 0, u0 , u1 , v0 , v1 such that 0 = z0 · · · zk−2 u0 , 1 = u1 zk · · · zm−1 , u0 u1 = zk−1 , 0 = z0 · · · zn−2 v0 , 1 = v1 zn · · · zm−1 and v0 v1 = zn−1 .

DECORATED LINEAR ORDER TYPES

9

Suppose k = n. We have u0 u1 = zk−1 = v0 v1 . So, we have either (a) u0 = v0 and u1 = v1 , or, for some w, either (b) u0 w = v0 and u1 = w v1 , or (c) u0 = v0 w and w u1 = v1 . In case (b), we have: 0 ∗ w = 0 and

1 = w ∗ 1 . We leave (a) and (c) to the reader. Suppose k < n. We have:

0 ∗ (u1 zk · · · zn−2 v0 ) = 0 and 1 = (u1 zk · · · zn−2 v0 ) ∗ 1 . The case that k > n is similar. It is easy to see that ⊕ is a sum or coproduct in the sense of category theory. The following theorem is immediate. Theorem 8. If a is an atom of Ai , then a is an atom of A0 ⊕ A1 . Finally, we have the following theorem. Theorem 9. Let A be any set and let B := B, ∗ be any concatenation structure. We assume that A and B are disjoint. Then, there is an extension of B with at least A as atoms. Proof. Let A∗ be the free semi-group generated by A. We can take as the desired extension of B, the structure A∗ ⊕ B. Remark 5.1. The whole development extends with only minor adaptations, when we replace axiom TC2 by: ˙ (∃!w (x ∗ w = u ∧ y = w ∗ v) ∨ • x ∗ y = u ∗ v → ((x = u ∧ y = v) ∨ ∃!w (x = u ∗ w ∧ y ∗ w = v)) ˙ is exclusive or. Here ∨ §6. A canonical construction. Although we know that not every concatenation structure can be represented by decorated linear orderings, i.e., as a concrete concatenation structure, there may exist a canonical construction of a concrete concatenation structure which is a representation whenever there exists any concrete representation. In this section we shall propose such a construction, but we can only show that it is universal in a restricted subclass of all concrete representations. It will be now more convenient to work with a theory for monoids, rather than for semigroups, as we did in the previous sections. We will work in the theory TCε+ , which is TCε0 plus the following axiom. TCε 9. x ∗ y ∗ z = y → (x = ε ∧ z = ε). We do not postulate the existence of irreducible elements, as they do not play any role in what follows, but they surely can be present. We shall call elements of a model M of TCε+ : words. When possible, the concatenation symbol ∗ will be omitted. When considering representations of structures with a unit element ε by decorated order types, one has to allow an empty decorated order structure.

10

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

Thus a representation of a model M of TCε+ is a mapping that assigns a decorated order structure to every w ∈ M so that 1. (ε) = ∅, 2. (uv) = (u)(v) and 3. (w) ∼ = (v) implies w = v. Lemma 10. In a model M of TCε+ the binary relation ∃u(xu = y) deﬁnes an ordering on the elements of M. Deﬁnition 6.1. A k-partition of a word w is a k-tuple (w1 , . . . , wk ) such that w1 . . . wk = w; we shall often abbreviate it by w1 . . . wk . An ordering relation is deﬁned on 3-partitions of w by: • u1 u2 u3 ≤ v1 v2 v3 :⇔ ∃x1 , x3 (v1 x1 = u1 ∧ x3 v3 = u3 ). The axioms ensure that for any two partitions there is a unique common reﬁnement. Deﬁnition 6.2 (Word Ultraﬁlters). Let w be a word and S a set of 3partitions of w. We shall call S a word ultraﬁlter on w if 1. εwε ∈ S 2. xεy ∈ S for any x, y 3. if U ∈ S, V is a 3-partition of w and U ≤ V , then V ∈ S 4. if xyz ∈ S and y = y1 y2 , then exactly one of the following two cases holds: (x, y1 , y2 z) ∈ S or (xy1 , y2 , z) ∈ S. Let S be a word ultraﬁlter on w and xyz ∈ S. Then we deﬁne the natural restriction of S to y which is a word ultraﬁlter Sy on y deﬁned by: • (r, s, t) ∈ Sy :⇔ (xr, s, tz) ∈ S. We shall deﬁne an ordering on word ultraﬁlters on a ﬁxed w and an equivalence on word ultraﬁlters on all words of M . Let S and T be word ultraﬁlters on w, then we deﬁne: • S < T :⇔ ∃u, v ((ε, u, v) ∈ S ∧ (u, v, ε) ∈ T ). Let S and T be word ultraﬁlters on possibly diﬀerent words, then we deﬁne: • S ∼ T :⇔ ∃x, x , y, z, z (xyz ∈ S ∧ x yz ∈ T ∧ Sy = Ty ). Notice that < is a strict ordering on word ultraﬁlters on w, but for S < T it still may be S ∼ T . Deﬁnition 6.3. Let w be a word. The canonical decorated ordering associated with w is the ordering of all word ultraﬁlters on w, where each word ultraﬁlter S is decorated by [S]∼ , the equivalence class of ∼ containing S. This decorated ordering will be denoted by C (w). Here are some basic properties of C (w). • The topological space determined by the ordering is compact and totally disconnected. In particular, it has largest and smallest elements.

DECORATED LINEAR ORDER TYPES

11

• For every proper preﬁx x of w, there is a uniquely determined pair of word ultraﬁlters which forms a gap (no word ultraﬁlter in between). Thus there is a natural embedding of the ordering of the preﬁxes into C (w). More precisely, we have have two mappings φw− and φw+ such that for a proper preﬁx x the pair φw− (x), φw+ (x) is the gap corresponding to x. If x = ε (or x = w), then only φw+ (x) (or φw− (x)) is deﬁned and it is the least (largest) element of C (w). Furthermore the images of these mappings are dense sets in C (w). • Vice versa, every gap in C (w) corresponds to a preﬁx (or equivalently to a 2-partition). • If a is an atom (irreducible element in M), then it determines a principal word ultraﬁlter. For a given atom a all such principal word ultraﬁlters are equivalent. C satisﬁes conditions 1. and 2. but, in general, condition 3. fails. Example 6.4. Let M = A∗ be the monoid generated by A (the alphabet). Then, all word ultraﬁlters are principal and C (w) is essentially the string w itself. Example 6.5. This is a ‘pathological example’. Let M be the nonnegative real numbers with +. For a positive real r, the order type of C (r) is the order type of: {(0, 1)}∪(I ×{0, 1})∪{(1, 0)}, where I is the open unit interval, with the lexicographic order. The equivalence relation ∼ has two classes; elements of the form (x, 0) are decorated by one type of word ultraﬁlters, elements (x, 1) are decorated by the other type. Hence for every r, s > 0, C (r) ∼ = C (s), thus C is not a representation. Deﬁnition 6.6. is a regular representation of M by decorated orderings, if for every 2-partition x1 x2 = w of w ∈ M, there exists a unique 2-partition A1 A2 = (w) such that A1 ∼ = (x1 ) and A2 ∼ = (x2 ). We do not know if every concatenation structure that has a concrete representation also has a concrete regular representation. If is regular, we have an analogous property for k-partitions for every k. For a k-partition (x1 , . . . , xk ) of w in M, we shall write k (x1 , . . . , xk ) = (A1 , . . . , Ak ), where (A1 , . . . , Ak ) is the uniquely determined k-partition of (x1 . . . xk ) such that Ai ∼ = (xi ), for i = 1, . . . , k. Theorem 11. 1. If the canonical mapping C is a representation of M, then it is a regular representation of M. 2. If there exists a regular representation of M, then so is also C . Proof. Ad 1. Let uv = w be in M, and suppose we have two diﬀerent 2partitions AB = C (w), A B = C (w), with A = C (u) ∼ = A , B = C (v) ∼ = B .

12

ˇ CI ˇ C, ´ P. PUDLAK, ´ V. CA G. RESTALL, A. URQUHART, AND A. VISSER

Suppose that A is a proper initial segment of A . Since AB corresponds to a 2-partition uv, there is a gap between A and B. Since A is a proper initial segment of A , the gap is in A . As every gap corresponds to a 2-partition of the preimage in C , there exists y and D, such that A = AD and D ∼ = C (y). Hence u = uy, which is possible only if y = ε. But then D is empty, which is a contradiction. Ad 2. Given a regular representation we should show condition 3. for mapping C . Since is a representation it suﬃces to show that C (u) ∼ = C (v) implies (u) ∼ = (v). Our strategy is (i) to construct, for every w ∈ M , an order preserving mapping h with h : Supp((w)) → Supp(C (w)), and then (ii) to show that if : C (u) → C (v) is an isomorphism, then for every S ∈ Supp(C (u)) the ﬁbers of S and (S), as decorated orderings, are isomorphic, i.e., h −1 (S) ∼ = h −1 ( (S)), or they are both empty. If (i) and (ii) are true, then it is easy to construct an isomorphism (u) ∼ = (v): it suﬃces to connect the isomorphisms of the ﬁbers into one isomorphism. Ad (i). Let w ∈ M , let j ∈ Supp((w)). We deﬁne: • h(j) := {(x, y, z) | ∃A, B, D (j ∈ B and 3 (x, y, z) = (A, B, D))}. One can readily verify that h(j) is a word ultraﬁlter, and that h is order preserving. Ad (ii). Let S ∈ Supp(C (u)) and T ∈ Supp(C (v)) such that T = (S). Then S and T have the same decoration, which means that S ∼ T . By deﬁnition, there exist 3-partitions (x, y, z) ∈ S and (x , y, z ) ∈ T such that Sy = Ty . Let 3 (x, y, z) = (A, B, D) and 3 (x , y, z ) = (A , B , D ). Then, B∼ = B , as is a representation. Let us denote this isomorphism by κ. Take an arbitrary 3-partition y1 y2 y3 = y and let 5 (x, y1 , y2 , y3 , z) = (A, B1 , B2 , B3 , D), and 5 (x , y1 , y2 , y3 , z ) = (A , B1 , B2 , B3 , D ). Then Bi ∼ = Bi , for i = 1, 2, 3. By the regularity of , the segments B1 , B2 , B3 in B and the segments B1 , B2 , B3 in B are uniquely determined by their isomorphism types, whence: (1)

κ(Bi ) = Bi , for i = 1, 2, 3.

The ﬁber h −1 (S) is deﬁned as the intersection of all segments B2 that belong to 5-partitions (x, y1 , y2 , y3 , z) such that (y1 , y2 , y3 ) ∈ Sy = Ty . Similarly, the ﬁber h −1 (T ) is deﬁned as the intersection of all segments B2 that belong to 5-partitions (x , y1 , y2 , y3 , z ) such that (y1 , y2 , y3 ) ∈ Sy = Ty . According to (1), for all such 3-partitions, κ : (B1 , B2 , B3 ) ∼ = (B1 , B2 , B3 ). Hence κ is also −1 −1 an isomorphism of h (S) onto h (T ), or both ﬁbers are empty.

DECORATED LINEAR ORDER TYPES

13

REFERENCES

[1] John Burgess, Fixing Frege, Princeton Monographs in Philosophy, Princeton University Press, Princeton, NJ, 2005. [2] Andrzej Grzegorczyk, Undecidability without arithmetization, Studia Logica. An International Journal for Symbolic Logic, vol. 79 (2005), no. 2, pp. 163–230. [3] Laurence Kirby, Addition and multiplication of sets, MLQ. Mathematical Logic Quarterly, vol. 53 (2007), no. 1, pp. 52–65. [4] P. Komjath ´ and V. Totik, Problems and Theorems in Classical Set Theory, Problem Books in Mathematics, Springer, New York, 2006. [5] W. V. Quine, Concatenation as a basis for arithmetic, The Journal of Symbolic Logic, vol. 11 (1946), pp. 105–114. [6] W. Sierpinski, Cardinal and Ordinal Numbers, Polska Akademia Nauk, Monograﬁe ´ ´ Matematyczne, vol. 34, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958. [7] A. Tarski, Der wahrheitsbegriﬀ in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1935), pp. 261– 405, Reprinted as [9]. The paper is a translation of the Polish Poje¸cie prawdy w je¸zykach nauk dedukcyjnych, Prace Towarzystwa Naukowego Warszawskiego, Wydział III matematyczno-ﬁzycznych, no. 34, Warsaw 1933. [8] , The notion of rank in axiomatic set theory a some of its applications. abstract 628, Bulletin of the American Mathematical Society, vol. 61 (1955), p. 433. [9] , The concept of truth in formalised languages, Logic, Semantics, Metamathematics (A. Tarski, editor), Oxford University Press, Oxford, 1956, (this paper is a translation of [7]), pp. 152–278. [10] A. Visser, Growing commas—a study of sequentiality and concatenation, Notre Dame Journal of Formal Logic, vol. 50 (2009), no. 1, pp. 61–85. DEPARTMENT OF MATHEMATICS ˇ UNIVERSITY OF ZAGREB, BIJENICKA 30 10000 ZAGREB, CROATIA

E-mail: [email protected] MATHEMATICAL INSTITUTE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC ˇ ´ 25, 115 67 PRAHA 1, CZECH REPUBLIC ZITN A

E-mail: [email protected] SCHOOL OF PHILOSOPHY, ANTHROPOLOGY AND SOCIAL INQUIRY THE UNIVERSITY OF MELBOURNE PARKVILLE 3010, AUSTRALIA

E-mail: [email protected] DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TORONTO, TORONTO ONTARIO, CANADA M5S 1A4

E-mail: [email protected] DEPARTMENT OF PHILOSOPHY UTRECHT UNIVERSITY, HEIDELBERGLAAN 8 3584 CS UTRECHT, THE NETHERLANDS

E-mail: [email protected]

CARDINAL PRESERVING ELEMENTARY EMBEDDINGS

´ EDUARDO CAICEDO ANDRES

Abstract. Say that an elementary embedding j : N → M is cardinal preserving if CARM = CARN = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V . We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M .

§1. Introduction. This paper is the ﬁrst of a series attempting to investigate the structure of (not necessarily ﬁne structural) inner models of the set theoretic universe under assumptions of two kinds: 1. Forcing axioms, holding either in the universe V of all sets or in both V and the inner model under study, and 2. Agreement between (some of) the cardinals of V and the cardinals of the inner model. I try to be as self-contained as is reasonably possible, given the technical nature of the problems under consideration. The notation is standard, as in Jech [8]. I assume familiarity with inner model theory; for ﬁne structural background and notation, the reader is urged to consult Steel [19] and Mitchell [15]. In the remainder of this introduction, I include some general observations on large cardinal theory, forcing axioms, and ﬁne structure, and state the main results of the paper. ¨ Consider set theory with the axiom of choice as formalized by the GodelBernays axioms GBC, so we can freely treat proper classes. An inner model (or simply, a model) is a transitive class model M of the Zermelo-Fraenkel ZFC axioms containing all the ordinals. If M is a model and ϕ is a statement, ϕ M is the assertion that ϕ holds in M . If is a deﬁnable term, M indicates the interpretation of inside M . Denote by ORD the class of ordinals and by CAR the class of cardinals. The coﬁnality of an ordinal α is denoted cf(α). All our embeddings are elementary and non-trivial, and the classes involved are inner models; the critical point of such an embedding j is denoted cp(j). Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

14

CARDINAL PRESERVING ELEMENTARY EMBEDDINGS

15

1.1. Large cardinal axioms. See Kanamori [10] as a general reference for large cardinals. It is well-known that the GBC axioms fall quite short of providing a complete picture of the universe of sets. Among the many independent extensions that have been studied, two sets of statements have been isolated as natural candidates to add to the basic axioms: large cardinal axioms and forcing axioms. By far, large cardinal axioms are better understood and more readily accepted. There is no formal deﬁnition of what a large cardinal is, but a few features can be distinguished. Typically, they are regular cardinals κ such that Vκ is itself a model of ZFC and, more importantly for present purposes, one can associate to κ a family of elementary embeddings j : V → M where M is a transitive class. The association is usually (as in the case of measurable, strong or supercompact cardinals) that κ is the critical point cp(j) of j, the ﬁrst ordinal α such that j(α) > α, but it can take other shapes, as is the case with Woodin cardinals. An important remark is that these notions can be stated in terms of the existence of certain ultraﬁlters or systems of ultraﬁlters called extenders, the connection being given by an analogue of the model theoretic ultrapower construction. An extender is essentially (a family of ultraﬁlters coding together) a fragment of an elementary embedding, and it is by now a standard device; good expositions and deﬁnitions can be found in Jech [8, Chapter 20], Kanamori [10, § 26], and Steel [19, § 2.1], among others. Brieﬂy: For a set X and a cardinal κ, let [X ]κ = {Y ⊆ X : |Y | = κ} and deﬁne [X ] κ. A non-trivial (κ, )extender E is a sequence (Ea : a ∈ [] n(. . . )’, so the computational challenge of the theorem is to ﬁnd a realizer or bound for ∃n. If we furthermore restrict the ε > 0 to those that can be written as 2−k for k ∈ N, we see that the theorem is almost purely arithmetical — except for parameters f and T which involve the types X . In a moment, we shall see how to cast even those parameters in a such a way that they only contribute to the computational challenge of the theorem by natural numbers. This theorem can be formalized in the formal system A [X, ·, ·]. The theory A is Peano arithmetic extended to all ﬁnite types and extended with

PROOF MINING IN PRACTICE

85

the axiom schema of dependent choice. Most of classical analysis can be formalized in this theory. The theory A [X, ·, ·] denotes the extension of A with an abstract Hilbert space in the following way: • We add a new ground type X for the elements of the Hilbert space and extend A to all ﬁnite types T X over N and X . • We add new constants representing the inner product and the vector space operations on X . • We add new axioms describing the algebraic properties of the inner product and the vector space operations. ¨ Using a monotone variant of Godel’s Dialectica interpretation, one proves general metatheorems about the extraction of bounds from proofs of ∀∃statements in A [X, ·, ·] and similar theories. For details, see [2]. The Mean Ergodic Theorem is a ∀∃∀-statement. As mentioned above, such theorems do not always allow to extract realizers or bounds for the existential quantiﬁer. This is also the case with the Mean Ergodic Theorem, where we can construct a Hilbert space and a mapping T such that a full rate of convergence for the ergodic averages An f would solve the halting problem for Turing machines. Even in the measure theoretic setting, one may construct counterexamples. Only when the measure space is ergodic, one obtains a full rate of convergence. For details, see [1]. Instead, we will extract bounds for the classically equivalent but constructively weaker no-counterexample version of the Mean Ergodic Theorem: Mean Ergodic Theorem (no-counterexample version). Let (X, ·, ·) be a Hilbert space, let T : X → X be a linear, nonexpansive selfmapping of n 1 i X . For f ∈ X deﬁne An f := n+1 i=0 T f, then for every ε > 0 and and every number-theoretic function M : N → N there exists an n ∈ N s.t. AM (n) f − An f < ε. The function M claims the existence of a counterexample to the convergence of An f, and the existence of n shows that any such supposed counterexample can be refuted. This no-counterexample version is classically equivalent to full convergence. Constructively, this establishes the local stability (up to an ε > 0) of the ergodic averages An f for arbitrarily long periods of time. The theorem is now in a suitable form for the metatheorems mentioned above to be applicable. From the most general form of the metatheorems in [2] we derive the following special case: Deﬁnition 2. The ﬁnite types T X are deﬁned as follows: (i) N, X ∈ T X , (ii) , ∈ T X ⇒ → ∈ T X . We often write 0 for the type N and 1 for the type N → N. Deﬁnition 3. A formula F is called a ∀-formula, resp. ∃-formula, if it is of the form ∀a Fqf , resp. ∃a Fqf , where Fqf is quantiﬁer-free and the types of

86

PHILIPP GERHARDY

are small1 , which includes amongst others the types N, X , N → N, N → X and X → X . We write F∀ and F∃ for ∀-formulas and ∃-formulas respectively. Corollary 4. Let (X, ·, ·) be an abstract Hilbert space. If ∀f X , T X →X , k 0 , M 1 T n.e. ∧ ∀u 0 B∀ (f, T, k, M, u) → ∃v 0 C∃ (f, T, k, M, v) is provable in A [X, ·, ·], then there is a computable φ : N × N × NN → N such that ∀f X , T X →X , k 0 , M 1 f ≤ b ∧ T n.e.∧ ∀u 0 ≤ φ(b, k, M )B∀ → ∃v 0 ≤ φ(b, k, M )C∃ holds in any Hilbert space (X, ·, ·). This is exactly the logical form of the no-counterexample version of the Mean Ergodic Theorem, and it is easy to see that the standard proof of the (no-counterexample version) Mean Ergodic Theorem can be formalized in A [X, ·, ·], as the proof only uses the basic algebraic properties of normed spaces and inner products. Thus, this corollary predicts a computable bound on ∃n in the Mean Ergodic Theorem in terms of (an integer bound b on) f, ε and the counterexample function M , but independent of the particular f (as long as its norm is bounded by b), the particular T or the particular space (X, ·, ·). In [1], the following bounds were obtained:

ik 2lb b nk = M i0 = 0, ε2 ε l =0 15 4 4 2 M (nk ) b ik + 1 = ik + ε4 2

Let d = 512b , then for some n ≤ N (b, ε, M ) = 2nεd b , we have that AM (n) f − ε2 An f < ε. In the next sections, we discuss details of the proof analysis carried out in [1] which resulted in these bounds. §3. Interpreting the law of the excluded middle. The main argument of the proof of the Mean Ergodic Theorem goes as follows: Any element f ∈ X can be written as a sum f = h + g, where h is T -invariant (i.e. Th = h) and g can be approximated arbitrarily well by an element of the form u − Tu with u ∈ X . Since T is linear, we may thus study An h and An g seperately. Obviously, An h = h for all n ∈ N. Also, An (u − Tu) ≤ u 2n , which quickly converges to 0. In other words, the main argument is: Either f −Tf = 0, f is T -invariant and An f is constant, or f −Tf > 0 and then An f converges, because the part that is not T -invariant tends to 0 quickly. To estimate how 1 For

the exact deﬁnition, see [2].

PROOF MINING IN PRACTICE

87

fast An (u − Tu) converges to 0, we need to know the norm u, or at least an upper bound on u. As the decomposition of f into g and h corresponds to a decompostion of the whole space X into orthogonal subspaces U (the closure of the space of elements of the form u − Tu) and V (the subspace of T -invariant elements), this is equivalent to approximating the projection of f onto U by some u − Tu and obtaining an upper bound on u. As the averages An f lie in the subspace generated by {f, Tf, T 2 f, . . . }, it suﬃces to consider the projection of f onto Uf = {T i f − T i+1 f|i = 0, 1, 2, . . . }. For this subspace Uf ⊆ U , we can explicitly deﬁne a sequence un such that gn = un − Tun converges to the projection of f onto Uf , and it is from this sequence that we derive an upper bound on the norms un . The elements un are deﬁned as follows: f, f − Tf f, f − Tf2 f − (ui − Tui ), T i+1 f − T i+2 f i+1 = T f. T i+1 f − T i+2 f2

u0 = ui+1

The numerator of the fraction has an easy upper bound by f, f − Tf ≤ ff−Tf, resp. f−(ui −Tui ), T i+1 f−T i+2 f ≤ fT i+1 f−T i+2 f. Thus we only need to ﬁnd a lower bound for f − Tf, resp. T i+1 f − T i+2 f in the denominator — but the norm of elements T i f − T i+1 f can be arbitrarily small. The solution lies in the original argument: If the norm of T i f − T i+1 f is zero for some i, then T i f is T -invariant and we are done. If not, it must be larger than some > 0. This is close to the solution, but not quite there yet. As we are only looking for local stability for An f, not full convergence, it suﬃces that T i f − T i+1 f is small enough, as then repeated use of the triangle inequality yields local stability. Given ε > 0 and the counterexample function M , we can say how small T i f − T i+1 f needs to be to directly imply local stability, i.e. we can produce an explicit > 0 such that T i f − T i+1 f < yields the result. Otherwise, we have a lower bound on T i f − T i+1 f and therefore an upper bound for ui ! In [1], we prove the following lemma (Lemma 2.15 in [1]), reﬂecting the above discussion: Lemma 5. For any i ≥ 0 and ε > 0, either 1. there is an n ≤ 2i f ε such that AM (n) f − An f ≤ ε, or f2 i f 2. ui ≤ 2ε j=0 M (2j ε ). In summary, we use the fact that for any i ∈ N either T i f − T i+1 f is zero or it is not zero. We give a computational interpretation of this instance of the law of the excluded middle by producing, for each i ∈ N, a > 0 such that both T i f − T i+1 f ≤ and T i f − T i+1 f > yield an n ∈ N such

88

PHILIPP GERHARDY

that AM (n) f − An f < ε. The maximum of the two is then an upper bound on an n ∈ N such that the result holds. §4. The principle of convergence for monotone sequences. Another important aspect of the proof is the use of the convergence of the sequence gn = un − Tun to the projection of f onto Uf . The convergence of gn is established in the following way: For every n ∈ N, the norm gn ≤ f. Also, gn ≤ gn+1 , so the sequence an = gn is a bounded monotone sequence of real numbers, and hence it is a Cauchy sequence. A simple argument shows that for every > 0 there is a > 0 such that if |ai − ai+d | < then gi − gi+d < , so that we can obtain a rate of convergence for gn from a rate of convergence for an . The convergence of gn is used to give an estimate of AM (n) f − An f. For any i, n ∈ N: AM (n) f − An f ≤ AM (n) (f − gi ) − An (f − gi ) + AM (n) gi + An gi . One then shows that there is > 0 such that if gi − gi+d < for all d > 0 then AM (n) (f − gi ) − An (f − gi ) < ε/2 for all n. With an upper bound on ui (such as the one we sketched in the previous section), we can ﬁnd an n large enough so that AM (n) gi , An gi < ε/4. Combined, this yields AM (n) f − An f < ε. Classically, the full convergence of the sequence gn is established using the principle of convergence for monotone sequences, but constructively, we cannot obtain a full rate of convergence for the gn , as the convergence of gn and the ∀∃∀ version of the Mean Ergodic Theorem are equivalent constructively. However, we may still give a computational interpretation of this particular appeal to the principle of convergence for monotone bounded sequences in the proof of the Mean Ergodic Theorem. As this may sound confusing, here is another explanation: We cannot obtain a computable rate of convergence, but we can give a computational interpretation to the use of its existence as a lemma. This is because a proof of a ∀∃-theorem cannot fully exploit the computational strength of a ∀∃∀-lemma. The principle of convergence for monotone bounded sequences is an instance of arithmetical comprehension. The computational interpretation of arithmetical comprehension in general requires bar recursion (see [6]). Even though comprehension is applied to a sequence an = gn which is explicitly given in the parameters of the proof, a computational interpretation of this instance of comprehension — although simpler — would still vastly increase the complexity of the extracted bounds. Instead we opt for a diﬀerent approach. In [3], Kohlenbach describes a technique to eliminate certain instances of arithmetical comprehension without increasing the growth rate (relative to the Grzegorczyk classes of computational complexity) of the bounds extracted by monotone functional interpretation. In this particular case, the informal

89

PROOF MINING IN PRACTICE

idea amounts to this: Although we obtain the Mean Ergodic Theorem from the full convergence of gn , we only need to establish gi − gi+d < for a particular d which is expressed in the other parameters and a hypothetical i. The exact d can be read oﬀ of the proof — as made explicit in the lemmas stated below — and from this we may form a sequence of non-overlapping intervals [ik , ik +dk ] where i0 = 0 and ik+1 = ik +dk . Now recall that there was a > 0 such that |ai − ai+d | < implies gi − gi+d < . Since the sequence an is monotone and bounded |aik − ajk | cannot exceed > 0 inﬁnitely often. Hence, |aik − ajk | < for some k ∈ N, and the result follows. In total, this amounts to replacing the use of full convergence with the statement that there is no counterexample to convergence, and the latter has, as sketched, a simple primitive recursive interpretation. Interestingly, the usefulness of this computationally weaker, no-counterexample version of convergence was recently (re-)discovered independently by Terrence Tao [7] under the name “Finite convergence principle”. Tao uses this principle to establish ﬁnitary, combinatorial proofs of some convergence results in ergodic theory, but without proving explicit bounds. See [8] for details. In [1], this argument yields the following sequence of lemmas, concerned with establishing AM (n) (f − gi ) − An (f − gi ) ≤ ε/2 by making gi − gi+d small enough for some i and d : 4

. Then for every i there is a j Lemma 6. Let ε > 0, let d = d (ε) = 32f ε4 in the interval [i, i + d ) such that T (f − gj ) − (f − gj ) ≤ ε. 4

4

Lemma 7. Let ε > 0, let n ≥ 1, let d = d (n, ε) = d (2ε/n) = 2n εf . 4 Then for every i there is a j in the interval [i, i + d ) such that An (f − gj ) − (f − gj ) ≤ ε. Lemma 8. Let ε > 0, let m ≥ 1, let d = d (m, ε) = d (m, ε/2) = 4 4 32mεf . Furthermore suppose gi − gi+d ≤ ε4 . Then for any n ≤ m, 4 An (f − gi ) − (f − gi ) ≤ ε. Lemma 9. Let ε > 0, let m ≥ 1, let d = d (m, ε) = d (m, ε/2) = 4 4 2 m εf . Furthermore suppose gi − gi+d ≤ ε8 . Then for any n ≤ m, 4 An (f − gi ) − Am (f − gi ) ≤ ε. 9

The last three lemmas follow fairly easy from the ﬁrst one, repeatedly appealing to the triangle inequality. The ﬁrst is proved using general properties of the inner product in Hilbert spaces; see [1] for details. The last lemma shows exactly, how the convergence of gn allows us to bound An (f−gi )−Am (f−gi ), where eventually m is replaced by M (n). In other words, this insight allows us to prove the above lemmas and later the no-counterexample convergence of An f already from the no-counterexample convergence of gn : The expression deﬁning dk in terms of ik and the other parameters is nothing but a counterexample function. In this case, there is

90

PHILIPP GERHARDY

a nested appeal to the no-counterexample convergence of gn , resp. an , as dk needs to be big enough enough to ensure that |adk − adk +1 | < k where k is given in terms of ik and the other parameters. Thus sk depends on ik , while ik depends on previous values dk−1 and thereby on ik−1 . The required > 0 such that gik − gik +dk < implies AM (n) f − An f ≤ ε, and thus the number of intervals [ik , ik + dk ] we need to consider, is independent of k though. Thus we consider longer and longer intervals [ik , ik + dk ], but only a ﬁxed number of those. For the details, see [1]. §5. Conclusions. In the previous three sections, we illustrated three aspects of a recent application of proof mining to the Mean Ergodic Theorem. At the heart of this proof analysis are proof theoretic ideas that originate from the general logical metatheorems in [2]. On the surface, this analysis is based on ideas not dependent of particular aspects of mathematical logic: (1) making the computational meaning or challenge of a theorem explicit, (2) giving a computational interpretation of an instance of the law of the excluded middle, and (3) reﬁning the appeal, within the proof, to the principle of convergence for monotone bounded sequences of real numbers. In this way, the author hopes that the above examples illustrate that proof mining yields analyses and reﬁnements of mathematical proofs that ought to be considered mathematical even by those who do not consider logic and proof theory proper mathematics. The proof theoretic perspective on mining proofs merely provides a systematic tool to carry out these analyses and to guide intuition where the result may not be obvious otherwise. Acknowledgements. The author would like to thank the reviewers for making helpful suggestions for improving the presentation in this paper.

REFERENCES

[1] J. Avigad, P. Gerhardy, and H. Towsner, Local stability of ergodic averages, to appear in Transactions of the American Mathematical Society. [2] P. Gerhardy and U. Kohlenbach, General logical metatheorems for functional analysis, Transactions of the American Mathematical Society, vol. 360 (2008), no. 5, pp. 2615–2660. [3] U. Kohlenbach, Elimination of Skolem functions for monotone formulas in analysis, Archive for Mathematical Logic, vol. 37 (1998), no. 5-6, pp. 363–390, Logic Colloquium ’95 (Haifa). [4] , Some logical metatheorems with applications in functional analysis, Transactions of the American Mathematical Society, vol. 357 (2005), no. 1, pp. 89–128. [5] , Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008. [6] C. Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proceedings of symposia in pure mathematics, AMS, Providence, 1962, pp. 1–27. [7] T. Tao, Soft analysis, hard analysis, and the ﬁnite convergence principle, On T. Tao’s blog: terrytao.wordpress.com.

PROOF MINING IN PRACTICE

91

[8] , Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynamical Systems, vol. 28 (2008), no. 2, pp. 657–688. [9] A. S. Troelstra (editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973. DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO BLINDERN, N-0316 OSLO, NORWAY

E-mail: [email protected]

CARDINAL STRUCTURE UNDER AD

STEVE JACKSON

Our aim in this paper is to survey the theory of cardinal structure assuming AD, the axiom of determinacy (deﬁned below). We work throughout in the theory ZF + DC + AD. The axiom of determinacy was introduced by Mycielski and Steinhaus in the early 60’s and quickly developed into a powerful tool for extending the ZF results of the classical descriptive set theorists about Σ11 (analytic) and Π11 (co-analytic) sets to higher levels of the projective hierarchy. The axiom contradicts the axiom of choice AC, but it was understood that it should be applied in a restricted universe such as L(R) (the smallest model of set theory containing the reals) where it seemed like a reasonable axiom. It wasn’t until much later, through the work of Martin, Steel, and Woodin (see [12], [20]), that it was shown that ZFC plus large cardinal assumptions actually imply that AD holds in L(R). The work of the Cabal from the late 60’s through the 80’s developed an extensive theory of pointclasses and associated properties from this axiom. A good reference for those developments is Moschovakis’ book [13] and the Cabal volumes themselves ([8], [9], [18], and [4]). This theory of the projective sets (and beyond) was largely developed in terms of certain ordinals called the projective ordinals, the 1n (Deﬁnition 1.12 below). It was also realized that AD had much to say about the properties of cardinals in general. An example would be Solovay’s early result that ℵn is singular with coﬁnality ℵ2 for n ≥ 3 (see Theorem 8.2 of [8]). The theory at the time, however, was not suﬃcient to calculate the values of the 1n for n ≥ 5 nor to develop the theory of the cardinals past ℵ . Kunen and Martin independently discovered the idea of a homogeneous tree. The precise concept and deﬁnition was formulated by Kechris. Using this concept, Kunen initiated a program for computing the 1n . The program stalled, however. Kechris’ article [8] gives an exposition of what was known at this time. In the early 80’s Martin proved a result which in current terminology showed the existence of the Martin tree. This generalized the earlier construction of the Kunen tree which played an important role in the AD theory of the ℵn . Building on this and some joint work with Martin, the author computed the 1n and developed 2000 Mathematics Subject Classiﬁcation. 03E60, 03E55, 03E05. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

92

CARDINAL STRUCTURE UNDER AD

93

a theory for analyzing the cardinal structure through the supremum of the 1n and a ways beyond. In particular, this theory of the projective ordinals generalizes in a straightforward manner to determine the cardinal structure through ℵ1 (the supremum of the 1n is ℵ0 , where 0 = supn (n) where we set (0) = 1 and (n + 1) = (n) ). In hindsight, a key concept that was missing in the earlier theory was the notion of a description. These are ﬁnitary objects which, roughly speaking, describe how to generate ordinals through the iterated ultrapowers of certain canonical measures. The earlier theory can in fact be viewed as a very simple case of the general theory, using only “trivial descriptions” (which are just integers). This point is explained in more detail in [3]. It turn out that the descriptions in fact completely describe the cardinal structure. One of the main goals of this paper is to introduce the descriptions in as simple a manner as possible, and to show how they determine the cardinal structure. In the complete analysis one must, in addition to deﬁning the descriptions, analyze the measures (countably additive ultraﬁlters) on the cardinals and establish certain partition properties (the strong partition property on the 12n+1 ) as well as verify certain inductive hypotheses. These additional arguments, however, use the same notion of description. In this paper, we focus on the descriptions themselves and the cardinal structure they generate. The reader wishing to see the details of the complete inductive analysis can consult [5] for the complete ﬁrst step of the inductive analysis (including the computation of 15 ) and [4] for the general step. The reader can also consult [3] for a more complete exposition of how descriptions are used in other ways (e.g., in analyzing the measures and establishing the partition properties). Knowledge of these other papers is not necessary for this paper, however. Rather than trying to explain in detail how descriptions are used in the complete inductive analysis, we introduce them here through two “exemplary problems.” These two problems can be stated entirely down at the level of the ℵn , but require the development of the same descriptions needed to compute 15 . As is turns out, 15 = ℵ +1 , and the descriptions we introduce through these two problems suﬃce to analyze the cardinal structure below this point. Through this approach, the reader can see clearly the underlying combinatorics in a much simpler setting. Our aim is that reader with a knowledge of basic determinacy theory at the level of [13] and perhaps [8] can follow the main thread of this paper. The reader could consult [10] or [13] for more general background on descriptive set theory. In §1 we give some background and sketch some of the “global” theory of AD. By this we mean the more general results that hold for all pointclasses and are independent of the more detailed analysis using the descriptions. This section is deﬁnitely of a survey nature and is included for the sake of completeness. In §2 we give a brief sketch of the overall plan of how the

94

STEVE JACKSON

inductive analysis of the projective ordinals goes. This section is also included for the sake of completeness and to give the reader a sense of how descriptions ﬁt into the bigger picture. The later sections do not speciﬁcally depend on these sections, and the reader wishing to quickly see the notion of description can skip them. In §3 we recall some basic deﬁnitions and facts about partition relations. Partition relations play a central role in how descriptions are used to generate ordinals. Our approach in this paper is to assume the strong partition property on the odd projective ordinals, the 12n+1 , and proceed straight to the descriptions. In the actual full analysis, the descriptions are also used to prove the partition relations. In §4 we introduce the two canonical families of measure W1m and S1m and the ﬁrst exemplary problem: compute the ultrapower jS1m (n ) of the cardinal n by the measure S1m . Solving this problem will only need the introduction of the “trivial” descriptions (actually slight generalizations of these). In §5 we consider the second exemplary problem: compute the iterated ultrapower jS m1 ◦ jS m2 ◦ · · · ◦ jS1mt (n ). The 1 1 answer is actually immediate from the solution to the ﬁrst problem, however we can describe the cardinal structure in the iterated ultrapower by introducing the next level of description. These are the same objects one uses in the full analysis to compute 15 , prove the strong partition relation on 13 , and the weak partition relation on 15 (which is the ﬁrst step of the inductive analysis). We carry out an example in computational detail. In §6 we sketch how these descriptions actually are used to generate the cardinal structure between 13 = ℵ+1 and 15 = ℵ . We also give a brief hint at what goes into the more general deﬁnition of description. In §7 we give an entirely diﬀerent mechanism which can be used to present the cardinal structure below the projective ordinals. This is done via a certain algebra which is deﬁned directly and does not need descriptions. Descriptions are needed, however, to prove that this alternate formulation works (these proofs are not given here). This ¨ [7] and extends earlier notational framework involves joint work with B. Lowe joint work of the author and F. Khaﬁzov [6]. Thus, although the proofs that this method works are quite involved, this gives a direct and self-contained mechanism for understanding the cardinal structure. This should make the techniques of this area accessible to a wider audience. Finally, in §8 we present as an application of the theory of cardinal structure under AD a result concerning the collapse of cardinals from L(R) to V . Namely, we sketch a proof of the fact that assuming large cardinals in V plus the saturation of the non-stationary ideal on 1 , some regular cardinal in L(R) below (ℵ2 )L(R) is collapsed in V . The proof we present here is only a sketch, as complete details will be given elsewhere. This section is not intended to be self-contained, and is included to give an example of how the theory of the cardinal structure under AD can be used to get results in the ZFC world.

95

CARDINAL STRUCTURE UNDER AD

§1. Basic concepts. Though our intention is to eventually specialize to results of a “local” nature, that is, pertaining to the smaller cardinals for which the inductive combinatorial methods work, we present as background here AD results of a more general nature. There results are “global” in nature in that they hold throughout the Wadge hierarchy under AD. The reader already comfortable with the basics of AD theory can skip this section, or skim it to see our (mostly standard) notation. Let denote the Baire space with the usual topology, i.e., the product of the discrete topology on . is homeomorphic to the set of irrationals (as a subspace of R), and as is customary, we often call the “reals.” We let Σ01 denote the collection of open sets, and Π01 the collection of closed sets in . The levels of the Borel hierarchy are deﬁned as usual by A ∈ Σ0α (for α < 1 ) iﬀ A = n∈ An where An ∈ Π0αn for some αn < α. Also, A ∈ Π0α iﬀ Ac ∈ Σ0α (we use Ac to denote − A). The collection of Borel sets is 0 1 α 0, and take M0 to be the real algebraic numbers. The family {(0, α ) : α ≡M0 α} has the ﬁnite intersection property but (0, α) ∩ M0 is empty. 3. In the stable case, the analogous theorem to Theorem 5.7 is true for any deﬁnable set because the assumption is equivalent to the forking of X (a) over M0 . 4. The description of a deﬁnably compact set using a type-deﬁnable open covering (see Fact 2.3) easily follows from Fact 5.7. End of Digression. Proof of Theorem 5.3. Because G is aﬃnely embedded it is closed and bounded in M k . Assume X ⊆ G is not left generic. By Fact 5.4 we may assume that X is closed. Fix M0 such that X is deﬁnable over M0 . By Fact 5.6, there exists g ∈ G such that M0 ∩ Xg = ∅. By Theorem 5.7, there are g1 , . . . , gk (each realizing the same type as g over M0 ) such that k i=1

Xgi = ∅.

PILLAY’S CONJECTURE

191

By taking complement we get k

(G \ X )gi = G.

i=1

Hence, G \ X is right-generic.

Remarks. 1. Theorem 5.3 fails without the deﬁnable compactness assumption: The set (a, +∞) and its complement are both not generic in R, + (here left and right-generic are the same). 2. The analogue of Theorem 5.3 in the stable setting is true for any deﬁnable subset of the group G. 3. Recently, Eleftheriou has pointed out that the assumption that G is aﬃnely embedded can be omitted Theorem 5.7 by working in the manifold charts of G. Here are two easy consequences: Fact 5.8. Assume that G is deﬁnably compact and abelian. (1) The non-generic sets form an ideal. (2) Every formula deﬁning a generic set in G belongs to a complete “generic” type p (over M). Namely, every formula in p deﬁnes a generic set in G. §6. Some theory of generic sets II: Measure and the NIP. The content of Sections 6-9 is mostly taken from [18]. The connection between the Non Independence Property and measure is due to Keisler [21] and the proof of Lemma 6.4 below is modeled after a proof from Keisler’s paper. The next notion is due to Shelah. The deﬁnition we use is equivalent to the original one. Deﬁnition 6.1. A theory T is said to be dependent or to have the Non Independence Property (NIP) if for every indiscernible sequence ai : i < over A and φ(x, y) a formula over A the type {φ(x, b2j ) φ(x, b2j+1 ) : j < } is inconsistent. (We take φ to mean (φ ∧ ¬) ∨ (¬φ ∧ )) Stable theories, o-minimal theories, the theory of p-adically closed ﬁelds all have the NIP, while the theory of pseudo-ﬁnite ﬁelds fails to have it. In fact, by Shelah’s work any simple unstable theory fails to have NIP. Deﬁnition 6.2. We say that G admits a left invariant Keisler measure if there exists a real valued ﬁnitely additive measure : Def(G) → R on the deﬁnable subsets of G, such that (G) = 1 and for every deﬁnable X ⊆ G and g ∈ G, we have (gX ) = (X ). In the rest of this section we make the following assumptions on the group G (equipped with the deﬁnable sets induced by the ambient structure):

192

YA’ACOV PETERZIL

• G has NIP. • The non left-generic sets form an ideal. • G admits a left-invariant Keisler measure. As we will eventually show, every deﬁnably compact group in an o-minimal structure satisﬁes all of the above. For now, notice that any abelian deﬁnably compact group satisﬁes the above assumptions. Indeed, o-minimality implies NIP, and by Fact 5.8 the non-generic sets form an ideal. Because every abelian group is amenable, it admits a left-invariant real valued ﬁnitely additive probability measure on all subsets. Deﬁnition 6.3. For X, Y ⊆ G deﬁnable, we write X ≈ng Y if X Y is not left-generic. Notice that since we assume that the non left-generics form an ideal ≈ng is an equivalence relation. The NIP assumption is crucial for the following. Lemma 6.4. The equivalence relation ≈ng is bounded. I.e., there exists a ﬁxed small model M0 such that every equivalence class of ≈ng is already represented by a deﬁnable set over M0 . Proof. Let denote the ﬁnitely additive left-invariant measure on Def(G), the family of deﬁnable subsets of G. Note that if X ⊆ G is a deﬁnable ngeneric set then we have (X ) 1/n. Assume that ≈ng is unbounded. Then there exists a formula φ(x, y) over the empty set, with the variable x for elements in G, and unboundedly many bi ’s, such that φ(G, bi ) φ(G, bj ) is generic. By standard Ramsey-type arguments, there exists a ﬁxed n and an indiscernible sequence ai : i < such that for every i = j, the set φ(G, ai ) φ(G, aj ) is n-generic. Consider the family F = {Yj = φ(G, a2j ) φ(G, a2j+1 ) : j < }. By indiscernibility, there exists a natural number k such that every k sets in F have empty intersection. However, for every j, (Yj ) 1/n, and because (G) = 1 it is impossible that every k sets in F intersect trivially. Contradiction. Deﬁnition 6.5. For X ⊆ G deﬁnable, let Stabng (X ) = {g ∈ G : gX ≈ng X }. Under our assumptions, the set Stabng (X ) is a subgroup of G. It is typedeﬁnable because for every n, the set of all g such that n translates of gX X do not cover G, is deﬁnable. The map g → gX/ ≈ng induces an injective map from G/Stabng (X ) into Def(G)/ ≈ng and therefore we proved: Theorem 6.6. For any deﬁnable X ⊆ G, the subgroup Stabng (X ) has bounded index in G. §7. The proof of PC in the abelian case. We assume in this section that M expands a real closed ﬁeld and that G is deﬁnably compact and abelian.

PILLAY’S CONJECTURE

193

Our goal here is to prove: Theorem 7.1. If G is deﬁnably compact and abelian then dimLie (G/G 00 ) = dimM G. Proof. Because G 0 has ﬁnite index in G we may assume that G is deﬁnably connected. The proof is based on two ingredients. The ﬁrst one is a deep theorem of Edmundo and Otero on the torsion points in deﬁnably connected, deﬁnably compact abelian groups. (Presumably, this was one of the main justiﬁcations to the original conjecture of Pillay). Its proof is based on Cohomological tools and uses extensively the triangulation theorem which is true only in o-minimal expansions of real closed ﬁelds: Theorem 7.2. [13] Assume that M expands a real closed ﬁeld and that G is a deﬁnably compact, deﬁnably connected abelian group of dimension n. Then Tor(G) Tor(Tn ). (where Tn is the n-dimensional torus and Tor(G) is the subgroup of torsion elements in G). The second ingredient, which we will prove below is: Lemma 7.3. G 00 is torsion-free. Let us see how the two results, taken together, imply PC in the abelian case: Lemma 7.3, together with the divisibility of G 00 (see Corollary 4.2 (1)) imply that Tor(G/G 00 ) Tor(G). If dim(G) = n, it follows from theorem [13], that Tor(G/G 00 ) Tor(Tn ). Because G/G 00 is a connected (Theorem 4.1) abelian compact Lie group, it is Lie isomorphic to a direct sum of T1 ’s. The number of these T1 ’s is determined by, say, the number of 2-torsion points, therefore G/G 00 (T1 )n and so the real dimension of G/G 00 is n. Proof of Lemma 7.3. For every n ∈ N, consider the map n : g → g n from G onto G. By deﬁnable choice, there exists a deﬁnable set X ⊆ G such that n |X is a bijection of X and G (we assume that G is deﬁnably connected). By [13] (or actually by the preceding results in [37]), Tn = ker( n ) is ﬁnite. It clearly contains all n-torsion points and, as easily checked, G equals a ﬁnite disjoint union of the gX ’s, for g ∈ Tn . Thus X and all the gX ’s are generic and pair-wise disjoint, and therefore Tn ∩ Stabng (X ) = {0}. Because this is true for every n, the group Stabng (X ) is torsion-free. By Theorem 6.6, Stabng (X ) has bounded index in G, and therefore G 00 ⊆ Stabng (X ). It follows that G 00 is torsion-free, ending the proof of Lemma 7.3 and thus PC in the abelian case.

194

YA’ACOV PETERZIL

§8. Proof of PC for arbitrary deﬁnably compact G. We assume in this section that G is a deﬁnably compact group in an o-minimal expansion of a real closed ﬁeld. Here are some preliminary facts about noncommutative deﬁnably compact groups: As shown in [31], G/Z(G) is semisimple, namely has no inﬁnite deﬁnable abelian normal subgroup. If we let N = Z(G)0 then the same is true for G/N . By [29], G/N can be written as an almost direct product of deﬁnably almost simple groups and every deﬁnably simple group is deﬁnably isomorphic to a semialgebraic linear group deﬁned over the real algebraic numbers. In particular, deﬁnably simple groups have very good reduction. The proof of PC for groups with very good reduction is partly contained in the Introduction (see [28] for more details). It easily follows that PC holds for deﬁnably compact almost simple groups and therefore also for an almost direct product of such groups. Therefore, PC holds for both N (Theorem 7.1) and for G/N . We thus have: dimM (G) = dimM (G/N ) + dimM (N ) = dimLie ((G/N )/(G/N )00 ) + dimLie (N/N 00 ). We also have dimLie (G/G 00 ) = dimLie (G/G 00 N ) + dimLie (G 00 N/G 00 ). It is easy to verify that G 00 N/N = (G/N )00 . Hence, (G/N )/(G/N )00 G/G 00 N. In order to show that dimM (G) = dimLie (G/G 00 ) it is therefore suﬃcient to prove: G 00 N/G 00 N/N 00 . The group on the left is isomorphic to N/(G 00 ∩ N ), hence in order to prove PC we are left to prove: Lemma 8.1. If G is deﬁnably compact then N 00 = G 00 ∩ N . The fact that N 00 ⊆ (G 00 ∩ N ) follows from the fact that the group on the right has bounded index in N . However, in order to prove the opposite inclusion (which fails for arbitrary groups, even with NIP) we need to take one more de’tour, through the general theory of generic sets. §9. Some theory of generic sets III. In this section we make no assumption on the group G unless otherwise stated.

PILLAY’S CONJECTURE

195

Deﬁnition 9.1. The theory of G is said to have ﬁnitely satisﬁable generics1 (in short fsg) if there exists a complete type p over M such that if φ(x) ∈ p then: (i) φ(G) is left generic. (ii) There exists a small model M0 ⊆ M such that every left translate of φ(G) intersects M0 . Our goal is to show that every deﬁnably compact group in an o-minimal structure has fsg. This is useful because of the following properties: Fact 9.2. Assume that T = Th(G) has fsg. Then (i) There exists a small M0 ⊆ M such that every left generic set and every right generic set intersect M0 . (ii) Given X ⊆ G deﬁnable, X is left-generic if and only if it is right generic. (iii) The deﬁnable non generic

sets in G form an ideal. (iv) G 00 exists and G 00 = {Stabng (X ) : X ∈ Def(G)}. Proof. Assume that p and M0 witness fsg. (i) If X is a left generic set then there are g1 , . . . , gk ∈ G such that the formula x = x is equivalent to the ﬁnite disjunction of the formulas x ∈ gi X . Hence, there is gi such that “x ∈ gi X ” is in p. By assumption on p, X ∩ M0 = ∅. Consider the type p(x −1 ). Because x → x −1 is a ∅-deﬁnable bijection of G it easily follows that if φ(x) ∈ p(x −1 ) then Y = φ(G) is right generic and every right translate of Y intersects M0 . As above, it follows that every right generic set intersects M0 . (ii) Assume X is not a left generic set. By Fact 5.6, there exists a right translate of X which does not intersect M0 , hence by (i), X is not right generic as well. (iii) Since p is a complete generic type it must contain the complement of every nongeneric set. (iv) For the existence of G 00 , see [18], Corollary 4.3. Now ﬁx a small model M0 witnessing (i). Given X ⊆ G deﬁnable, let M1 be a small model containing M0 over which X is deﬁnable. If g ≡M1 h then gX ∩ M1 = hX ∩ M1 and hence (gX hX ) ∩ M0 = ∅. By (i), gX hX is nongeneric. Thus, every coset of Stabng (X ) contains all the realizations of some complete type over M1 . In particular, in a (still small) model where every complete type over M1 is realized, there is a representative for every coset of Stabng (X ), so Stabng (X ) has bounded index, and therefore it contains G 00 . For the opposite inclusion, since G 00 has bounded index it can be obtained as the intersection of deﬁnable generic sets (Fact 5.5). If g belongs to Stabng (X ) for every such X then it must belong to G 00 (otherwise, by compactness, there is X containing G 00 such that gX ∩ X = ∅, which implies that g∈ / Stabng (X ). 1 For

simplicity, we slightly modiﬁed the deﬁnition from [18].

196

YA’ACOV PETERZIL

Lemma 9.3. Assume that N is a deﬁnable normal subgroup of G. If N and G/N have fsg then G has fsg. Proof. See Proposition 4.5 in [18]. We return to the o-minimal setting. Lemma 9.4. If G is deﬁnably compact and abelian in an o-minimal structure then G has fsg. Proof. Since we do have complete generic types in abelian groups (see Fact 5.8), it is suﬃcient to show that there is a small M0 such that every generic set intersects M0 . Let M0 be a small model such that every ≈ng -class in Def(G) has a representative deﬁnable over M0 (recall, Lemma 6.4, that ≈ng is bounded because G has NIP). Given X ⊆ G generic, there exists X1 ⊆ X such that X1 is still generic and Cl (X1 ) ⊆ X . Indeed, the following argument for that fact was suggested by the UIUC Logic seminar (it assumes that M expands an ordered group but this is unnecessary, as the argument in [18] shows): G=

k

gi X.

i=1

By Fact 5.4, Int(X ) is also generic. For every > 0 let X = {g ∈ X : d (g, Fr(X )) > } (the notion of d (g, Fr(X )), the distance of g from the frontier of X , assumes the presence of an underlying group). We have, X . X = >0

It is now suﬃcient to take realizing the complete type p(x) of the inﬁnitesimals right of zero. So, G=

k

gi X .

p i=1

We obtained a deﬁnable open covering of G parameterized by a complete type. By the equivalent deﬁnition to deﬁnable compactness, Fact 2.3, there is a ﬁnite subcover, which easily implies that some X is generic. If we let X1 = X then Cl (X1 ) ⊆ X . We may therefore assume that X is closed. Let Y ⊆ G be a set deﬁnable over M0 such that Y X is nongeneric (the existence of Y follows from our assumption on M0 ). Because X is generic so is Y . Again, by Fact 5.4, we may assume that Y is closed, so both X and Y are deﬁnably compact. We will show that (Y ∩ X ) ∩ M0 = ∅ and in particular X ∩ M0 = ∅. Notice that both X and Y are deﬁnably compact.

PILLAY’S CONJECTURE

197

By Theorem 5.7, if X ∩ Y ∩ M0 = ∅ then there are ﬁnitely many M0 conjugates of X ∩ Y whose intersection is empty. Because Y is M0 -deﬁnable there are X1 , . . . , Xk all M0 -conjugates of X such that k

Xi ∩ Y = ∅.

i=1

Since Y is generic this implies that for some Xi we must have Y \ Xi generic. Contradiction to Y X being non-generic. Lemma 9.5. If H is deﬁnably compact and semisimple in an o-minimal structure then H has fsg. The proof of this lemma is based on the almost-decomposition into deﬁnably almost simple groups. The deﬁnably simple case is handled in [28] using measure theoretic arguments based on [7] and [1]. Using Lemmas 9.4, 9.5, and 9.3, we can conclude: Theorem 9.6. Every deﬁnably compact group in an o-minimal structure has fsg. The above theorem, together with Fact 9.2, implies that the set of left (hence also right) generics in G form an ideal, and that for any deﬁnable set X , Stabng (X ) is a type-deﬁnable group of bounded index. Finally (and this is the main fact which forced us to take this de‘tour through the notion of “fsg”), the group G 00 is the intersection of all stabilizers of deﬁnable subsets of G. §10. Completing the proof of PC. We can now return to the missing ingredient in the proof of PC, namely the proof of Lemma 8.1. We need to show that G 00 ∩ N = N 00 , where N is a deﬁnably connected normal central subgroup. By Corollary 4.2, it is suﬃcient to prove that G 00 ∩N is torsion-free. Given n ∈ N, let Tn = Torn (N ) and X ⊆ N be a deﬁnable set such that g → g n gives a bijection of X and N . By Deﬁnable Choice, there is D ⊆ G which intersects every coset of N exactly once. It is now easy to verify that G is the ﬁnite disjoint union of the translates of DX by the elements of Tn . In particular, DX is generic and Tn ∩ Stabng (DX ) = {e}. Because this is true for every n, we have Stabng (DX ) ∩ Tor(G) = ∅. By the fsg property, G 00 ⊆ Stabng (DX ), therefore G 00 ∩ Tor(N ) = {e}. We thus proved that G 00 ∩ N = N 00 , completing the proof of PC (see the argument preceding Lemma 8.1). 10.1. Deﬁning measure on G. As a result of the work on Pillay’s Conjecture, the following theorem was established in [18].

198

YA’ACOV PETERZIL

Theorem 10.1. If G is deﬁnably compact in an o-minimal structure then it admits a left-invariant Keisler measure on the deﬁnable subsets of G. For a deﬁnable X ⊆ G, we have (X ) = 0 if and only if X is non-generic. Proof. As we already pointed out, the existence of such measure is immediate when G is abelian. In the general case, we ﬁrst note that G/G 00 , as a compact Lie group, admits a left-invariant ﬁnitely additive probability measure on a boolean algebra of sets containing all Borel measurable sets- the Haar measure m. We ﬁrst ﬁx a complete generic type p(x) over G. Given a deﬁnable set X , we consider the set Xˆ = {gG 00 ∈ G/G 00 : p |= “x ∈ gX }. (note that X is well deﬁned. Namely, if gh −1 ∈ G 00 then in particular, gX hX is non-generic and therefore not in p. It follows that gX ∈ p if and only if hX ∈ p). The main part of the proof is to show that Xˆ is a Borel set in G/G 00 (see Proposition 6.2 in [18]). We then deﬁne p (X ) = m(Xˆ ). Clearly, p is left invariant, and it is easy to verify that it is also ﬁnitely additive (if X1 ∩ X2 = ∅ then Xˆ1 ∩ Xˆ2 = ∅). Finally, if X is generic then ﬁnite additivity implies that p (X ) > 0 and if X is non-generic then Xˆ = ∅ and therefore p (X ) = 0. §11. Related work and some open questions. This section has gone through substantial changes in the last stages of writing. As will be explained below, most of the open questions listed here were solved in a recent paper by Hrushovski and Pillay, [20]. 11.1. Omitting the real closed ﬁeld assumption. As was pointed out early on, the only remaining obstacle for proving PC without the assumption that M expands a real closed ﬁeld is the lack of an analogue to Theorem 7.2 on the number of torsion points, without the ﬁeld assumption. Such a theorem was proved by Eleftheriou and Starchenko [16] when M was assumed to be an ordered division ring over an ordered vector space and hence PC holds in this case as well. Actually, a very clear description of deﬁnable groups in this setting is given in the paper, out of which the number of torsion points is easily read. In order to prove the torsion points result under weaker assumptions it seems important to develop similar topological tools to the ones originally used, but this time without the triangulation theorem. Indeed, Sheaf Cohomology in expansions of ordered groups has been the subject of several papers of Edmundo, Jones and Peatﬁeld (see [11] and [12]) and of Beraducci and Fornasiero (see [4]). In [14], Edmundo and Terzo prove Pillay’s conjecture

PILLAY’S CONJECTURE

199

under a relatively weak assumption on M but with an additional assumption of “orientability” on the group G. In a recent result, [27], I was able to prove the question about the torsion points and hence Pillay’s Conjecture, in o-minimal expansions of ordered groups, follows. The questions formulated below were written prior to the publication of the recent pre-print by Hrushovski and Pillay [20]. As I will eventually point out, most of these questions are now solved by that paper, either explicitly or implicitly. I leave them here because I ﬁnd that their discussion could still be of some interest. 11.2. Uniform deﬁnability of G 00 . An important feature of the basic example of Pillay’s conjecture (where we start with a compact real Lie group and view it in an elementary extension) is the fact that the type deﬁning G 00 is given by a single formula, with varying parameters. Namely, G 00 = {g ∈ G : |g| < a : a ∈ R}. Consider the structure Gind whose universe is G/G 00 , with a function symbol for the group operation and a predicate for every set of the form (X ), for X ⊆ G n deﬁnable in the o-minimal structure M. In [18] we showed, using a theorem of Baysalov and Poizat, that if G = [0, 1)n , + mod 1 (in an ominimal expansion of an ordered divisible abelian group) then structure Gind is deﬁnable in an o-minimal structure over the reals. Later, in [22], Marikova reproved this result without referring to [1], and provided a much ﬁner analysis of the deﬁnable sets in this structure. The uniformity in parameters plays an important role in both works. For more recent work of Marikova and v.d. Dries see [23], [39] Conjecture. If G is deﬁnably compact then there is a formula φ(x, y), where x varies over elements of G, and a set of parameters A, such that φ(g, a) . G 00 = g ∈ G : a∈A

Conjecture. The structure on the compact Lie group Gind is deﬁnable in some o-minimal structure over the real numbers. Related to the above conjecture is the following: Question. What is the structure which G induces on Tor(G)? In particular, what subsets of Tor(G) are of the form X ∩ Tor(G) for a deﬁnable subset of G? Note that when G is abelian its torsion group can be realized as a deﬁnable set in the o-minimal structure Q, +, 0. Proof. Existence and uniqueness of H is clear. Assume there are B, ε as stated. Then K := B \ int{F = 0} is compact and G does not have zeroes on K . Let c ∈ R, such that c ≥ ε1 and c1 ≤ |G|K |. Then for every x ∈ Rn we have |H (x)| ≤ c · |F (x)|: this holds true if F (x) = 0, since F = H · G and H vanishes on {G = 0}. If F (x) = 0, then x ∈ B or x ∈ B \ int{F = 0} = K . In both cases we get the assertion by the choice of c. 3.3. Corollary. Let A be bounded super real closed. Let r ∈ R, F1 , F2 ∈ C pol (Rn ) and a1 , . . . , an ∈ A be such that |ai | ≤ r (1 ≤ i ≤ n) and such that

BOUNDED SUPER REAL CLOSED RINGS

225

F1 (x) = F2 (x) for all x in an open set containing [−r, r]n ⊆ Rn . Then F1,A (a1 , . . . , an ) = F2,A (a1 , . . . , an ). n Proof. Deﬁne G : Rn −→ R by G(x) = i=1 sup{0, |xi | − ri }. Then G ∈ C (Rn ) and {G = 0} = [−r, r]n ⊆ int{F1 − F2 = 0} by assumption. By 3.2, there is some H ∈ C (Rn ) with F1 − F2 = H · G and since G ≥ 1 outside [−r − 1, r + 1]n we know that |H | ≤ c · |F1 − F2 | for some c ∈ R. Since F1 , F2 are polynomially bounded, also H ∈ C pol (Rn ). Thus F1,A (a1 , . . . , an ) − F2,A (a1 , . . . , an ) = HA (a1 , . . . , an ) · GA (a1 , . . . , an ). Since n|ai | ≤ r for each i and A is a real closed ring we know that GA (a1 , . . . , an ) = i=1 sup{0, |ai | − ri } = 0, which implies the corollary. 3.4. Proposition and Deﬁnition. Let A be a bounded super real closed ring. The holomorphy ring Hol A is a bounded super real closed subring of A and there is a unique super real closed ring structure on Hol A, which expands the bounded super real closed ring structure. For F ∈ C (Rn ) and a1 , . . . , an ∈ Hol(A) we have (†)

FHol A (a1 , . . . , an ) = GHol A (a1 , . . . , an )

whenever G ∈ C pol (Rn ) is such that for some r ∈ N with |ai | ≤ r we have F (x) = G(x) (x ∈ Rn , |x| ≤ r + 1). Proof. Hol A is a bounded super real closed subring of A, since for all a1 , . . . , an ∈ Hol A and each F ∈ C pol (Rn ), there are r ∈ R with |ai | ≤ r and a bounded F ∗ ∈ C ∗ (Rn ) such that F (x) = F ∗ (x) (|x| ≤ r + 1); hence by 3.3, FA (a1 , . . . , an ) = FA∗ (a1 , . . . , an ) ∈ Hol A. By 3.3, we may use (†) to deﬁne an LΥ -structure on Hol(A) which by deﬁnition expands the bounded super real closed ring structure on Hol A. It is straightforward (using 3.3) to check that this deﬁnes the unique super real closed ring structure on Hol A which expands the bounded super real closed ring structure. pol n 3.5. Theorem. Let F ∈ C (R ) and P(T ) ∈ R[T ], T = (T1 , . . . , Tn ) of total degree d with |F | ≤ |P| on Rn . Then there is a unique continuous function G ∈ C (Rn × R) with (∗)

¯ y) ∈ Rn × R). F (x1 , . . . , xn ) · y d +1 = G(x1 · y, . . . , xn · y, y) ((x,

Moreover G is polynomially bounded. Proof. Existence is given by [Tr2, (7.2)](ii). Uniqueness holds, since ¯ y) ∈ Rn × R with G(x1 , . . . , xn , y) is uniquely determined by (∗) for all (x, y = 0. Moreover the proof of [Tr2, (7.2)](ii) shows that G is again polynomially bounded (by a polynomial of total degree d . 3.6. Theorem. Let A be a bounded super real closed ring and let 1 ∈ S ⊆ A be multiplicatively closed. Then there is a unique expansion of the localization

226

MARCUS TRESSL

S −1 A to a bounded super real closed ring such that the localization map A −→ S −1 A is a bounded super homomorphism. The operation of F ∈ C pol (Rn ) on (S −1 A)n is given as follows: Pick d ∈ N0 such that F is bounded by a polynomial of total degree d and take a polynomially bounded continuous function G ∈ C (Rn × R) with F (x1 , . . . , xn ) · y d +1 = G(x1 · y, . . . , xn · y, y) ((x, ¯ y) ∈ Rn × R). Such functions exist by 3.5. Then for f1 , . . . , fn ∈ A and g ∈ S fn GA (f1 , . . . , fn , g) f1 ,..., := ∈ S −1 A. FS −1 A g g g d +1 Proof. The proof is parallel to the proof of the localization theorem [Tr2, (7.4)], using 3.5 instead of [Tr2, (7.2)](i). 3.7. Corollary. Let ϕ : A −→ B be a super homomorphism between bounded super real closed rings and let 1 ∈ S ⊆ A be multiplicatively closed such that ϕ(S) ⊆ B × . Then the natural map S −1 A −→ B is a super homomorphism, too. Proof. This follows immediately from the explicit deﬁnition of the bounded super real closed structure on S −1 · A in 3.6. §4. The super real closed hull. For a bounded super real closed ring A, we shall now deﬁne the smallest super real closed ring containing A as a bounded super real closed subring. 4.1. Theorem and Deﬁnition. Let A be a bounded super real closed ring. Let Aˆ = S −1 · Hol A, where S is the closure of A× ∩ Hol A under multiplication and Υ (recall: this means “closed under all the functions sHol A , s ∈ Υ”); here we consider Hol(A) equipped with the super real closed ring structure deﬁned in 3.4. Then there is a unique LΥ -structure on Aˆ such that Aˆ is a super real closed ring having A as a bounded super real closed subring. Aˆ is called the super real closed hull of A. Proof. Firstly, as A× ∩ Hol A ⊆ S we have A = (A× ∩ Hol A)−1 · Hol A ⊆ −1 ˆ By 3.4, Hol(A) is a bounded super real closed subring of S · Hol A = A. A and there is a unique expansion of this structure to a super real closed ring. By deﬁnition, S is closed under multiplication and Υ. By 3.1, there is a unique LΥ -structure on Aˆ such that Aˆ is a super real closed ring having Hol A as a super real closed subring. Since Aˆ is also the localization of A at S, 3.6 implies ˆ that A is a bounded super real closed subring of A. ˆ It remains to show that A with the LΥ -structure deﬁned above is the unique super real closed ring structure on Aˆ having A as a bounded super real closed subring. However, any other super real closed ring B expanding the pure

BOUNDED SUPER REAL CLOSED RINGS

227

ring Aˆ having A as a bounded super real closed subring, has Hol A as a super real closed subring (cf. [Tr2, (9.2)](i)) and the underlying bounded super real closed ring structure is the one induced from A. By 3.4, the super real closed ring structures of B and Aˆ induced on Hol A are equal. From the uniqueness ˆ property in 3.1 we know that B is the super real closed ring A. 4.2. Corollary. Let F, G ∈ C pol (Rn ). ¯ = 0 → G(x) ¯ = 0. (i) If {F = 0} ⊆ {G = 0}, then TΥpol ∀x¯ F (x) ¯ ≥ 0 → G(x) ¯ ≥ 0. (ii) If {F ≥ 0} ⊆ {G ≥ 0}, then TΥpol ∀x¯ F (x) Proof. (i). Let A |= TΥpol . By [Tr2, (5.5)](iv) the super real closed ring Aˆ is a model of ¯ = 0 → G(x) ¯ = 0. ∀x¯ F (x) Since A is a bounded super real closed subring of Aˆ (by 4.1), also A is a model of this sentence. (ii) follows from (i), since in every real closed ring A, the formula x ≥ 0 is equivalent to pA (x) = 0, where p : R −→ R is the inﬁmum of the identity function and the constant function 0. 4.3. Lemma. Let A be a bounded super real closed subring of the super real closed ring B. There is a unique A-algebra homomorphism Aˆ −→ B and this homomorphism is an embedding of super real closed rings. Proof. We have S0 := A× ∩ Hol A ⊆ T := B × ∩ Hol B. Since B is super real closed, T is closed under Υ: this follows from [Tr2, (6.12)], which says that all maximal ideal of B are Υ-radical. Since Hol A is a super real closed subring of Hol B by 3.4, T ∩ Hol A is Υ-closed as well. Thus the closure S of S0 under Υ and multiplication is contained in T , too. Hence we get a unique A-algebra homomorphism ϕ : Aˆ = S −1 · Hol(A) −→ T −1 · Hol(B) = B and this map is injective. It remains to show that ϕ is a super homomorphism. This follows immediately from the deﬁnition of the LΥ -structure on both rings in 3.1. 4.4. Corollary. If A is a super real closed ring, then Aˆ (deﬁned for the underlying bounded super real closed ring) is equal to A. In particular, the LΥ -structure of A is uniquely determined by the LΥpol -structure. 4.5. Corollary. Let B be a super real closed ring and let A be a bounded super real closed subring of A. Then B ∼ =A Aˆ as (bounded ) super real closed rings if and only if B is generated by A as a super real closed ring. Proof. Let C ⊆ Aˆ be the super real closed subring generated by A. By 4.3 there is a super real A-algebra monomorphism Aˆ −→ C . Composing this ˆ map with the inclusion C −→ Aˆ and using uniqueness shows that C = A. ˆ Hence A is generated by A as a super real closed ring.

228

MARCUS TRESSL

Conversely suppose B is generated by A as a super real closed ring. By 4.3, we may view Aˆ as a super real closed subring of B. Since B is generated by A ˆ we get B = A. 4.6. Proposition. Let A be a bounded super real closed ring. Then A is ˆ in other words Aˆ is a subring of the convex closure B of A. There convex in A, is a unique super real closed ring-structure on B extending the bounded super real closed ring structure on A. In particular, every bounded super real closed ring which is convexly closed (e.g. a ﬁeld ) has a unique expansion to a super real closed ring. Proof. Since Hol(A) is convex in A, the convex closure B of Hol A contains A. By [Tr2, (11.2)](iii) we know that B is the localization of Hol A at the set T of all non zero-divisors t of Hol A with the property that Hol A is convex in (Hol A)t . It follows A× ∩ Hol A ⊆ T . Since T is closed and closed under Υ by [Tr2, (11.11)], the closure S of A× ∩ Hol A is contained in T . Hence ˆ Aˆ = S −1 · Hol(A) ⊆ T −1 · Hol(A) = B, in other words A is convex in A. By [Tr2, (11.12)], there is a (unique) expansion of B to a super real closed ring having Aˆ as super real closed subring. Since B is a localization of Aˆ we get the uniqueness statement of the proposition from the uniqueness statement in 3.6 together with 4.4. 4.7. Corollary. Let A be a bounded super real closed subring of a super real closed ring B. Then A is convex in the super real closed ring generated by A in B. Proof. By 4.5 and 4.6. §5. Super real ideals. 5.1. Deﬁnition. An ideal I of a bounded super real closed ring A is called super real if sA (I ) ⊆ I for every s ∈ Υpol . Observe that in this case I is a radical ideal, in particular I is convex and satisﬁes a ∈ I ⇐⇒ |a| ∈ I (a ∈ A). Certainly,√ every ideal I of A is contained in a smallest super real ideal of A, denoted by Υ I . If A is a super real closed ring, then by [Tr2, (6.10)], the super real ideals are precisely the Υ-radical ideals (clearly Υpol is a set of generalized root functions as deﬁned in [Tr2, (3.2)]). 5.2. Examples. Let A be a bounded super real closed ring. (i) If F ∈ C pol (R) is strictly positive everywhere, then in general FA (a) is not a unit for every unit a ∈ A. For example if A is the bounded super real closed ring C pol (R), F = exp(−x 2 ) and a = 1 + x 2 ∈ A. (ii) If a ∈ A is a unit, then in general, there is some s ∈ Υpol , which is bounded away from 0 outside [−1, 1] such that sA (a) is not a unit. For

BOUNDED SUPER REAL CLOSED RINGS

229

1 ) example if A is the bounded super real closed ring C pol (R), s = exp(− |x| 1 and a = 1+x 2 ∈ A. Hence in this example, the ideal I = (s(a)) of A is proper, but the super real radical of I is not proper. In particular, maximal ideals of bounded super real closed rings are not super real in general. The example also shows that this is not resolved if we replace Υpol by the set of all s ∈ Υpol , which are bounded away from 0 outside a neighborhood of 0: or to replace Υpol by the set of all bounded s ∈ Υ such that s does not have zeroes diﬀerent from 0 in the Stone-Cech compactiﬁcation R of R.

5.3. Remark. If A ⊆ B is an extension of rings and I is an ideal of A, then I · B denotes the ideal generated by I in B. Recall that for a convex subring A of a real closed ring B and every radical ideal I of A we have I · B = {a · b | a ∈ I, b ∈ B} and this ideal is again radical. Our ﬁrst goal in this section is to show that 5.3 remains valid in the bounded super real closed context. That is, whenever A ⊆ B is a convex extension of bounded super real closed rings and I is a super real ideal of A, then I · B is a super real ideal of B (cf. 5.7). In order to prove this we show that for every s ∈ Υpol , there are t ∈ Υpol and F ∈ C pol (R2 ) with s(x · y) = t(x) · F (x, y). This is achieved in 5.6 below. First a preparational lemma: First two preparational lemmas from elementary analysis: 5.4. Lemma. Let A ⊆ R2 be compact with projection [a, b] onto the ﬁrst coordinate. Let C be the convex hull of A. Then C is again compact and the function f : [a, b] −→ R deﬁned by f(x) = max Cx is continuous, concave (i.e. f(x + (1 − )y) ≥ f(x) + (1 − )f(y) for all 0 ≤ ≤ 1) and satisﬁes f(a) = max Aa , f(b) = max Ab . Here Cx denotes the set {y ∈ R | (x, y) ∈ C } and similarly for Aa , Ab . Moreover, if A is the graph of a strictly increasing function [a, b] −→ R, then also f is strictly increasing.

Proof. This is clear.

5.5. Lemma. Let sn ∈ Υ (n ∈ N). Then there is some t ∈ Υ, 0 ≤ t ≤ 1, symmetric (i.e. t(−x) = t(x)), non-decreasing and concave in [0, ∞) such that for every n ∈ N there is some > 0 with t(x) ≥ |sn (x)| (|x| < ). Proof. We may assume that all sn are symmetric, non-decreasing on [0, ∞) and 0 ≤ sn ≤ 12 . Deﬁne 1 |x| 1 , + sup min , sn (x) (x) = min 2 2 n n

230

MARCUS TRESSL

which is continuous on R, strictly increasing in [0, 1] with values in [0, 1], (0) = 0 such that for every n, there exists > 0 with (x) ≥ sn (x) for 0 ≤ x ≤ . We deﬁne a function t : [0, 1] −→ [0, 1] as follows: Let C be the convex hull of the graph of [0, 1] and let t(x) := sup Cx (0 ≤ x ≤ 1), where Cx = {y ∈ R | (x, y) ∈ C }. By 5.4, t is a strictly increasing and concave homeomorphism [0, 1] −→ [0, (1)]. We extend t to R via t(x) = t(1) if x ≥ 1 and t(x) = t(−x) if x < 0. Then t is symmetric, 0 ≤ t ≤ 1 and it is clear that t is still concave in [0, ∞). Since t(x) ≥ (x) for all x ∈ [0, 1] it is clear that for all n ∈ N, there is some > 0 with t(x) ≥ sn (x) (|x| < ). 5.6. Proposition. Let s ∈ Υ. There are t ∈ Υ with 0 ≤ t ≤ 1, c ∈ R and F ∈ C (R2 ) such that s(x · y) = t(x) · F (x, y) and |F (x, y)| ≤ c · 1 + (1 + |y|) · |s(x · y)| ((x, y) ∈ R2 ) Proof. Let s0 (x) = x and for n > 0, sn (x) := max|y|≤n n · |s(y · x)|. Then sn ∈ Υ and from 5.5 we get some t ∈ Υ, symmetric with 0 ≤ t ≤ 1, nondecreasing and concave in [0, ∞) such that for every n ∈ N0 there is some > 0 with t(x) ≥ |sn (x)| (|x| < ). By deﬁnition of sn for n ≥ 1 this means (∗)

t(x) ≥ n · |s(yx)| (|x| < , |y| ≤ n).

We ﬁrst show that the function s(x·y) , deﬁned on (R\{0})×R has a continuous t(x) extension F through 0 on R × R: | < n1 for all Pick b ∈ R. For n ∈ N we have to ﬁnd some > 0 with | s(x·y) t(x) x ∈ (−, ), x = 0 and all y with |b − y| < . Enlarge n if necessary such that |b| < n and take > 0 with |b| + < n such that (∗) holds. Let 0 < |x| < and |b − y| < . Then |y| < |b| + < n. Thus |s(xy)| ≤ n1 t(x), as desired. It remains to ﬁnd c ∈ R such that for all (x, y) ∈ R2 , x = 0 we have s(x · y) (†) t(x) ≤ c · (1 + (1 + |y|) · |s(x · y)|). By choice of t there is some > 0 such that t(x) ≥ |s(x)| and t(x) ≥ |x| for all x with |x| < . It is enough to ﬁnd an element c satisfying (†) separately on each of the following four subsets of R2 , covering R2 : Case 1. |x| ≥ .

BOUNDED SUPER REAL CLOSED RINGS

231

Then t(x) = t(|x|) ≥ t() > 0 since t is symmetric and increasing in [0, ∞). 1 | ≤ | s(x·y) | and we may choose c := t() . Hence | s(x·y) t(x) t() Case 2. |x| < and |y| ≤ 1. As F is continuous we may choose c as the maximum of |F | on the rectangle [−, ] × [−1, 1]. Case 3. |x| < and |y| ≥ 1 and |x · y| ≥ . Then by the choice of we have t(x) = t(|x|) ≥ |x|, hence | s(x·y) | ≤ t(x) s(x·y) | s(x·y) x | ≤ |y · |, since 1 choose c = .

1 |x|

≤ | y | by assumption in case 3. Hence we may

Case 4. |x| < and |y| ≥ 1 and |x · y| < . 1 1 ≤ 1 we have t(x) = t(|x|) = t( |y| · |x · Since t is concave in [0, ∞) and |y| 1 1 y|) ≥ |y| · t(|x · y|) = |y| · t(x · y). Since |x · y| < we have t(x · y) ≥ |s(x · y)| by the choice of . Hence s(x · y) t(x · y) t(x) ≤ |y| · t(x · y) = |y| and we may choose c = 1.

5.7. Proposition. Let A be a convex subring of a bounded super real closed ring B. If I is a super real ideal of A then I · B is super real, too. Proof. For a ∈ I , b ∈ B and s ∈ Υpol we have to show that sB (a ·b) ∈ I ·B. By 5.6 there are t ∈ Υpol , c > 0 and F ∈ C (R2 ) with s(x · y) = t(x) · F (x, y) ((x, y) ∈ R2 ) such that |F (x, y)| ≤ c · (1 + (1 + |y|) · |s(x · y)|) everywhere. Since s is polynomially bounded also F is polynomially bounded. Hence sB (a · b) = tB (a) · FB (a, b). Since tB (a) = tA (a) ∈ I we get the claim. If A is a super real closed ring and I is an ideal of A, then there is a largest super real ideal I Υ of A contained in I and I Υ = {a ∈ I | sA (a) ∈ I for all s ∈ Υ}. (cf. [Tr2, (6.7)]). With the aid of 5.7, this can be extended to bounded super real closed rings: 5.8. Proposition and Deﬁnition. Let A be a bounded super real closed ring. If I is an ideal of A, then there is a largest super real ideal I Υ contained in I . We have I Υ = {a ∈ I | sA (a) ∈ I for all s ∈ Υpol } = (I ∩ Hol A)Υ · A. Proof. Let J := (I ∩ Hol A)Υ . By 5.7 we know that J · A is super real. Moreover it is clear that every super real ideal of A contained in I has to be contained in K := {a ∈ I | sA (a) ∈ I for all s ∈ Υpol }. In particular J · A ⊆ K and it remains to show that K ⊆ J · A.

232

MARCUS TRESSL

1 a a Pick a ∈ K . Since 1+a 2 , 1+a 2 ∈ Hol A we have 1+a 2 ∈ I ∩ Hol A and it a a Υ remains to show that 1+a 2 ∈ (I ∩ Hol A) . It suﬃces to show sA (| 1+a 2 |) ∈ I for every strictly increasing s ∈ Υ and indeed by [Tr2, (6.7)] it suﬃces to take √ √ a a | ≤ |a| we have s (| |) ≤ s (|a|) ∈ I by our s ∈ Υpol . Since | 1+a 2 A 1+a 2 A 1 choice √ of a in K . Now the aconvexity condition fora real closed rings implies that s A (|a|) divides sA (| 1+a 2 |) in A. Hence sA (| 1+a 2 |) ∈ I as desired. 5.9. Theorem. Let I be an ideal of a bounded super real closed ring A. Then √ √ Υ Υ Υ Υ I · Aˆ = I · Aˆ and I = I · Aˆ ∩ A. √ √ Υ Υ Proof. The inclusionΥ I · Aˆ ⊆ I · Aˆ follows from Υ I ⊆ I · Aˆ and √ √ Υ Υ I · Aˆ ⊇ I · Aˆ holds, since by 5.7, Υ I · Aˆ is super real. the inclusion √ √ Υ Υ ˆ ∩ A contains I and it remains to show that Clearly ( I · A) √ √ Υ ˆ ∩ A ⊆ Υ I. ( I · A) √ ˆ ∩ A. In order to show b ∈ I We may assume that I = Υ I . Take b ∈ (I · A) 2 we may replace b by b , hence we may assume that b ≥ 0. Since 1 + b 2 is a b b 2 ˆ unit in A we have 1+b 2 ∈ (I · A) ∩ A. Since b = 1+b 2 · (1 + b ) we may replace b ˆ b with 1+b 2 and we may assume that 0 ≤ b ≤ 1. Since b ∈ I · A, there are ˆ a ∈ I and c ∈ A with b = a · c. As b ≥ 0, b = |b| = |c| · |a| and we may assume that a, c ≥ 0, too (observe that I is radical, hence |a| ∈ I ). By 4.5, Aˆ is generated by A as a super real closed ring. Thus there are F ∈ C (Rn ) and a1 , . . . , an ∈ A with c = FAˆ(a1 , . . . , an ). Pick ϕ : [0, ∞) −→ [1, ∞) continuous and strictly increasing with |F (x)| ¯ ≤ ϕ(|x|) ¯ (x¯ ∈ Rn ). Deﬁne t : R −→ R by ⎧ |y| ⎪ ⎨ if y = 0 1 t(y) = ϕ( y 2 ) ⎪ ⎩ 0 if y = 0.

Using [Tr2, (7.2)](i) with s(x) = x we get t ∈ Υ and some G ∈ C (Rn × R) with (∗)

F (x1 , . . . , xn ) · t(y) = G(x1 y, . . . , xn y, y) on Rn × R.

Since ϕ is strictly increasing and ≥ 1 everywhere it is straightforward to see that t|[0.∞) : [0.∞) −→ [0.∞) is an homeomorphism which is polynomially bounded and whose compositional inverse is polynomially bounded, too. Hence t ∈ Υpol and there is some t1 ∈ Υpol with t ◦ t1 (y) = t1 ◦ t(y) = y for all y ≥ 0. As a ≥ 0 we get a = tA (t1,A (a)) from 4.2(ii). Since I is a super real ideal, also a0 := t1,A (a) ∈ I . From (∗) we then get b = c · a = FAˆ(a1 , . . . , an ) · tAˆ(a0 ) = GAˆ(a1 · a0 , . . . , an · a0 , a0 ). 1 The

convexity condition says: 0 ≤ a ≤ b ⇒ b|a 2 .

BOUNDED SUPER REAL CLOSED RINGS

233

Let H := (G ∨ 0) ∧ 1. Then in R we have ∀(x, ¯ y) : 0 ≤ G(x1 y, . . . , xn y, y) ≤ 1 ⇒ G(x1 y, . . . , xn y, y) = H (x1 y, . . . , xn y, y). Since this sentence is also valid in Aˆ and 0 ≤ b ≤ 1 we get b = HAˆ(a1 · a0 , . . . , an · a0 , a0 ). Since H is bounded it follows b = HA (a1 · a0 , . . . , an · a0 , a0 ). As H (0) = 2 ) for G(0) = 0, there is some s ∈ Υpol with H (z1 , . . . , zn+1 ) ≤ s(z12 + · · · + zn+1 2 all z1 , . . . , zn+1 ∈ R: choose s so that s(t) ≥ max{H (z) | zi ≤ t} (t ≥ 0). It follows 0 ≤ b = HA (a1 · a0 , . . . , an · a0 , a0 ) ≤ sA ((a1 a0 )2 + · · · + (an a0 )2 + 2 a0 ). Since a0 ∈ I and I is super real, we get b ∈ I as desired. Note that in general for a proper ideal I of a bounded super real closed ring ˆ Υ (e.g. if I · Aˆ = A, ˆ cf. 5.2(ii)) A, the ideal I Υ · Aˆ is properly contained in (I · A) 5.10. Scholium. Let A be bounded super real closed ring. An ideal of A is super real if and only if I is the kernel of a bounded super homomorphism A −→ B into a bounded super real closed ring. Proof. If ϕ : A −→ B is such a homomorphism and a ∈ I , then sA (a) ∈ I , since ϕ(sA (a)) = sB (ϕ(a)) = sB (0) = 0. Conversely suppose I is super real. By 5.7, I · Aˆ is super real, too. Together with 5.9 it follows that I ·Aˆ is a super real ideal of Aˆ lying over I . By [Tr2, (6.3)], super real ideals of Aˆ are kernels of super homomorphisms. Hence we can ˆ · Aˆ and we get that I is the kernel of a compose A −→ Aˆ with Aˆ −→ A/I bounded super homomorphism. 5.11. Corollary. Let A be bounded super real closed and let I ⊆ A be a super real ideal. There is a unique LΥpol -structure on A/I such that A/I is a bounded super real closed ring and the residue map A −→ A/I is a bounded super real homomorphism.

and this Moreover, there is a unique A-algebra homomorphism Aˆ −→ A/I ˆ homomorphism is super real with kernel I · A. In particular, there is a unique A-algebra isomorphism of super real closed rings ∼ = ˆ ˆ −→ A/I A/(I · A)

Proof. By 5.7 we know that I · Aˆ is a super real ideal of Aˆ lying over I . Since super real ideals are kernels of super homomorphisms by 5.10, we can ˆ · Aˆ and get that I is the kernel of a bounded compose A −→ Aˆ with Aˆ −→ A/I super homomorphism. The image is A/I and it is clear that the LΥpol -structure on A/I is uniquely determined by saying that the residue map A −→ A/I is a bounded super real homomorphism. ˆ · A, ˆ which is a bounded super We get an embedding of rings A/I −→ A/I

real homomorphism. By 4.3, we may view A/I as a super real closed subring

234

MARCUS TRESSL

ˆ · A. ˆ Since Aˆ is generated by A as a super real closed ring, also A/I ˆ · Aˆ of A/I

ˆ · Aˆ = A/I . Hence is generated by A/I as a super real closed ring, thus A/I

ˆ ˆ There we have a super real homomorphism ϕ : A −→ A/I with kernel I · A. ˆ can only be one such A-algebra homomorphism, since A is the localization of ˆ × ∩ A. A at (A) 5.12. Theorem. If ϕ : A −→ B is a bounded super homomorphism between bounded super real closed rings A and B, then there is a unique extension of ϕ to a ring homomorphism ϕˆ : Aˆ −→ Bˆ and this extension is super real. The functor F from bounded super real closed rings to super real closed rings, which maps A to Aˆ and ϕ to ϕˆ is an idempotent mono-reﬂector. This means: F is left adjoint to the inclusion from the category of super real closed rings into the category of bounded super real closed rings, F ◦ F = F and the adjoint morphism A −→ Aˆ is a monomorphism. Proof. First we prove the assertion about ϕ. Uniqueness again follows ˆ × ∩ A. Existence of ϕˆ from the fact that Aˆ is the localization of A at (A) follows from 5.11 and 4.3. Hence the functor F is well deﬁned. By 4.4, F ◦ F = F , which also shows that F is a reﬂector. F is a mono-reﬂector, since A −→ Aˆ is a monomorphism. ˆ We conclude this section by showing that the reﬂector A → A is also wellbehaved with respect to localization: 5.13. Proposition. Let A be a bounded super real closed ring, let 1 ∈ S ⊆ A be multiplicatively closed and let T be the closure of S in Aˆ under multiplication and Υ. Recall from 3.1 that there is a unique super real closed ring structure on T −1 · Aˆ such that the localization map Aˆ −→ T −1 · Aˆ is a super homomorphism. −1 · A induced by the localization map The natural morphism ϕˆ : Aˆ −→ S −1 · A)× and the induced map ϕ : A −→ S −1 · A, sends T into (S −1 · A T −1 · Aˆ −→ S

is an A-algebra isomorphism of super real closed rings. Proof. Since ϕ(S) consists of units of S −1 · A also ϕ(S) ˆ consists of units −1 of S · A. Since T is the closure of S under multiplication and Υ, ϕ(T ˆ ) is the closure −1 · A is super real closed, every of ϕ(S) ˆ under multiplication and Υ. Since S −1 · A is super real (cf. [Tr2, (6.12)]), hence for every for maximal ideal of S −1 · A, b is a unit in S −1 · A if and only if every s ∈ Υ and each element b ∈ S −1 −1 · A)× . s(b) is a unit in S · A. This proves that indeed ϕ(T ˆ ) ⊆ (S −1 · A is an isomorIn order to show that the induced map T −1 · Aˆ −→ S phism it now suﬃces to verify the universal condition deﬁning T −1 · Aˆ in the

BOUNDED SUPER REAL CLOSED RINGS

235

−1 · A, more precisely for the morcategory of super real closed rings for S −1 · A. Let : Aˆ −→ B be a super homomorphism into a phism Aˆ −→ S super real closed ring B with (T ) ⊆ B × . Then A : A −→ B is a super homomorphism with |A (S) ⊆ B × and by 3.7 there is a unique super homo−1 · A −→ B morphism h : S −1 ·A −→ B such that |A = h ◦ϕ. By 5.12, hˆ : S ˆ is the unique super homomorphism extending h with = h ◦ ϕ. ˆ

§6. The super real core. 6.1. Proposition. Let A0 be a convex subring of the super real closed ring A. Then there is a largest super real closed subring of A that is contained in A0 . Proof. By [Tr2, (9.2)](i), the convex hull of a super real closed subring of A is itself a super real closed subring of A. Hence, by using Zorn, it is enough to show for convex super real closed subrings B, C of A, that the ring D generated by B and C in A is again a super real closed subring of A. By [Tr2, (10.5)] we know that D is a convex subring of A and by [Tr2, (9.2)](i), it is enough to show that D is closed under Υ: Let b1 , . . . , bn ∈ B and c1 , . . . , cn ∈ C . Pick s ∈ Υ. It is enough to show |sA (b1 c1 + · · · + bn cn )| ≤ d for some d ∈ D. We may certainly assume that s is symmetric (i.e. s(−x) = s(x)) and strictly increasing on (0, ∞). The Cauchy-Schwarz inequality implies s(x1 y1 + · · · + xn yn ) ≤ s(|x||y|) for all x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈ Rn , where |x|, |y| denote the euclidean norm of x, y respectively. Since s(|x||y|) ≤ s(|x|2 ) + s(|y|2 ) we get s(x1 y1 + · · · + xn yn ) ≤ s(|x|2 ) + s(|y|2 ) on Rn × Rn . Thus sA (b1 c1 + · · · + bn cn ) ≤ sA (b12 + · · · + bn2 ) + sA (c12 + · · · + cn2 ) ∈ D as desired. 6.2. Corollary and Deﬁnition. For any bounded super real closed ring A there is a largest bounded super real closed subring, denoted by AΥ with the Υ = AΥ . We call AΥ the super real core of A. property A ˆ which is Proof. By 6.1, AΥ is the largest super real closed subring of A, contained in A. Observe that AΥ is convex in A, since Hol A ⊆ AΥ . For a proper ideal I of A we know I Υ = (I ∩ Hol A)Υ · A from 5.8. Hence I Υ = (I ∩ AΥ )Υ · A as well. √ √ Υ On the other hand Υ I ∩ AΥ in general properly contains I ∩ AΥ (e.g. if √ Υ I = A). 6.3. Corollary. For any bounded super real closed ring A we have AΥ = {a ∈ A | sAˆ(a) ∈ A for all s ∈ Υ}. Proof. Since AΥ is a super real closed subring of Aˆ we have “⊆”. Conversely take a ∈ A with sAˆ(a) ∈ A for all s ∈ Υ. Let B be the super real closed ˆ Thus B = {F ˆ(a) | F ∈ C (R)}. Certainly every subring generated by a in A. A element of B is bounded in absolute value by some sAˆ(a) for some s ∈ Υ.

236

MARCUS TRESSL

Hence by choice of a, the convex hull C of B in Aˆ is contained in A. C is a super real closed subring of B by [Tr2, (9.2)](i). Hence a ∈ C ⊆ AΥ . 6.4. Observation. If B is a real closed ring and A ⊆ B is a convex subring, then A is a domain if and only if B is a domain, and A is local if and only if B is local (as follows from the Gelfand-Kolmogorov Theorem). In particular for every bounded super real closed ring A we have (i) A is a domain ⇐⇒ Aˆ is a domain ⇐⇒ AΥ is a domain ⇐⇒ Hol A is a domain. (ii) A is local ⇐⇒ Aˆ is local ⇐⇒ AΥ is local ⇐⇒ Hol A is local. 6.5. Examples. Let A := {f ∈ C (R2 ) | f is polynomially bounded in the second coordinate}. Hence f ∈ A if and only if for every x ∈ R, the function f(x, ) : R −→ R is polynomially bounded. Clearly A is a convex subring of C (R2 ), hence A is a bounded super real closed subring of C (R2 ). We have Aˆ = C (R2 ) and AΥ = {f ∈ C (R2 ) | f is bounded in the second coordinate}. Here a super real closed ring properly between C ∗ (R) and C (R): Take A = {f ∈ C (R) | f is bounded on (0, ∞)}. Also note that there are many super real closed ring properly between C ∗ ([0, ∞)) and C ([0, ∞)), e.g. A = {f ∈ C (R) | f is bounded on N} has this property since x · distN (x) ∈ A \ C ∗ (R). The formation of the super real core is functorial: If ϕ : A −→ B is a bounded super homomorphism between bounded super real closed rings, then ϕ|AΥ is a super homomorphism AΥ −→ B Υ : since ϕˆ respects the LΥ structure on Aˆ by 5.12, ϕ(AΥ ) is a super real closed subring of Bˆ contained in B, i.e. ϕ(AΥ ) ⊆ B Υ . Hence the assignment A −→ AΥ is functorial, by sending ϕ to ϕ|AΥ . We shall not make use of this here. Instead, we state another description of the super real core. Since Hol A ⊆ AΥ ⊆ A, there are subsets S of Hol A with AΥ = S −1 ·Hol A. We can compute the largest such set upon input A: 6.6. Proposition. For any bounded super real closed ring A, the largest multiplicatively closed subset S of Hol A satisfying AΥ = S −1 · Hol A is S = {a ∈ Hol A | sHol A (a) ∈ A× for all s ∈ Υ}. Proof. The super real closed subrings of Aˆ contained in A are all of the form T −1 · Hol A, where T ⊆ A× ∩ Hol A. Since AΥ is the largest super real

BOUNDED SUPER REAL CLOSED RINGS

237

closed subring of Aˆ contained in A, the set T := (AΥ )× ∩ Hol A is the largest among all of them. It remains to show T = S. If a ∈ T , then a is a unit in AΥ and since AΥ is super real closed, all elements sAΥ (a) are units of AΥ as well. Since (AΥ )× ⊆ A× we get a ∈ S. Conversely let a ∈ S. The set T0 := {sHol A (a) | s ∈ Υ} is closed under multiplication and Υ (note that s1 , s2 ∈ Υ implies s1 (x) · s2 (x) ∈ Υ and s1 ◦ s2 ∈ Υ). Therefore T0−1 · Hol A has a unique super real closed ring ˆ Since a ∈ S we know T0 ⊆ A× and therefore structure (induced from A). −1 T0 · Hol A ⊆ A. So by the choice of T we obtain T0 ⊆ T . Thus a ∈ T0 ⊆ T . REFERENCES

[Schw] N. Schwartz, The basic theory of real closed spaces, Memoirs of the American Mathematical Society, vol. 77 (1989), no. 397, pp. viii+122. [Tr1] M. Tressl, Computation of the z-radical in C (X ), Advances in Geometry, vol. 6 (2006), no. 1, pp. 139–175. [Tr2] , Super real closed rings, Fundamenta Mathematicae, vol. 194 (2007), no. 2, pp. 121–177. UNIVERSITY OF MANCHESTER SCHOOL OF MATHEMATICS OXFORD ROAD, MANCHESTER M13 9PL, UK

E-mail: [email protected]

ANALYTIC COMBINATORICS OF THE TRANSFINITE: A UNIFYING TAUBERIAN PERSPECTIVE

ANDREAS WEIERMANN

Abstract. From a Tauberian perspective we prove and survey several results about the analytic combinatorics of (transﬁnite) proof-theoretic ordinals. In particular we show how certain theorems of Petrogradsky, Karamata, Kohlbecker, Parameswaran, and Wagner can be used to give a uniﬁed treatment of asymptotics for count functions for ordinals. This uniform approach indicates that (Tauberian theorems for) Laplace transforms provide a general tool to establish connections between additive and multiplicative results and may therefore be seen as a contribution to Problem 12.21 in Burris’s book on number theoretic density and logical limit laws. In the last section we give applications and prove in some detail phase transitions related to Friedman style combinatorial well-orderdness principles for fragments of ﬁrst order Peano arithmetic.

§1. Introduction to the transﬁnite combinatorics of the transﬁnite. Some years ago the author had a discussion with another logician about which ﬁelds in mathematics are surely have no connection with each other. A suggestion was: The theory of transﬁnite ordinal numbers and complex analysis. Of course this seems a safe guess since it is diﬃcult to imagine that Cauchy’s integral formula has something to say about the ordinals below or below ε0 . It has therefore been surprising that those connections exist and give rise to interesting cross-disciplinary applications. Perhaps even more interestingly we apply in this paper results from a paper in Lie-algebra theory to questions in logic. When the author gave a talk about these developments he has moreover been asked whether there are possible connections between ordinals and Tauberian theory, in particular Karamata’s theorem. The questioner was of course convinced that no natural connections exist. But the opposite is true. Karamata’s theorem is of considerable importance for studying Ackermannian functions (when resulting from < -descent recursive functions). Moreover in this article we show how classical Tauberian theorems can be used for studying the provably recursive functions for the fragments of ﬁrst order Peano arithmetic. Logic Colloquium ’07 Edited by Franc¸oise Delon, Ulrich Kohlenbach, Penelope Maddy, and Frank Stephan Lecture Notes in Logic, 35 c 2010, Association for Symbolic Logic

238

ANALYTIC COMBINATORICS OF THE TRANSFINITE

239

Historically the subject analytic combinatorics of the transﬁnite emerged from investigations of the author about Friedman style independence results for PA [35]. For these investigations asymptotic results on the counting of ordinals of a prescribed complexity turned out to be of crucial importance. Quite a few logicians would be content with the insight that these complexities are bounded exponentially but for the purpose of classifying phase ¨ transitions for Godel incompleteness results better bounds are needed. This is exactly the place where complex analysis and Tauberian theorems come into play. Cauchy’s formula provides bounds for the coeﬃcients of the generating function for the objects in question. (Similarly Tauberian theory, for example, via Karamata’s or Ingham’s theorem can be used for related purposes.) To be more speciﬁc let for < ε0 the number N denote the number of occurrences of in the Cantor normal form for . Let cα (n) := |{ < α : N = n}|. For small values of α the function cα can be evaluated by hands. For α = d already some calculation (see, for example, Theorem 2.48 in [8]) is needed to verify that c d (n) ∼

1 n d −1 . d !(d − 1)!

If one goes to α = the result is even more exciting, since we obtain √ 2n e 3 (1) c (n) ∼ √ . 4 3n (Here and in the sequel we denote by ∼ asymptotic equivalence. Thus an ∼ bn means as usual limn→∞ abnn = 1.) Now one can make a principal decision. Either one does not like such formulae at all or one get’s excited and wants to know how to obtain and how to apply them. (A simple proof of the formula (1) by induction seems ¨ managed to provide an elementary hopeless at least at ﬁrst. But still Erdos proof of (1) in [13]. We would also would like to mention that a Tauberian proof of (1) has been given in [19].) Mathematical experience from the last century (and even before) suggests that attacking diﬃcult counting problems is best done by studying related generating functions. Odlyzko commented for example in [24]:“Analytic methods are extremely powerful and when they apply, they often yield estimates of unparalleled precision.” Therefore let ∞ cα (n)z n . Cα (z) := n=0

240

ANDREAS WEIERMANN

Then under some standard conditions for r < 1 it holds that 1 Cα (z) (2) dz. cα (n) = 2i |z|=r z n+1 By comparing the coeﬃcients of the involved power series we see that (3)

C (z) =

∞ i=1

1 1 − zi

where C (z) is (uniformly compact) converging for |z| < 1. By exploiting (2) and (3) Hardy and Ramanujan (already in 1917) proved the formula (1) by introducing the circle method. In fact they proved a much stronger result which later was brought into perfection by Rademacher (see, for example, [3] for an accessible presentation) who provided a closed expression for c (n).

⎞ ⎛ 1 2 1 − sinh ∞ k 3 24 1 √ d ⎜ ⎟ kAk (n) ⎜ c (n) = √ ⎟ ⎝ ⎠ d 1 2 k=0 − 24 =n where Ak (n) =

h,k e −

2hn k

0 0. d +1 (1− )κd 1 . Since Let f(s) = (1 − )κ s d and h(v) = v d U (u) = exp (1 + o(1)) · u d +1 , d

we can choose a subset of S ⊆ {α < } such that cS (≤ v) ∼

exp(vh(v) + f(h(v))) h(v) 2f (h(v))

as v → ∞ (where f denotes the second derivative of f and cS (≤ v) the number of elements in S with norm not exceeding v. Note that the term on the right hand side is exactly of the form of equation (21.7) in [22].) Then Laplace’s method (cf. exercise 208 on page 80 in [30]) yields

∞ ∞ 1 −su −sn (5) e cS (≤ u)du = cS (n)e ∼ exp (1 − )κ d s· s 0 n=0

as s → 0 (where cS (≤ v) denotes the number of elements in S with norm equal to v). Let M (S) be the set of ﬁnite (normal form) sums (multisets) of elements d so that c d (n) ≥ from S. Then M (S) is contained in α : α < cM (S) (n). So it suﬃces to get a good lower bound on cM (S) (n). Let CS (s) :=

∞

cS (n)e −sn ,

n=0 ∞ ∞ 1 CS (sm) H (s) := m m=1

n=0

∞

and assume that H (s) = n=0 c (n)e −sn . Then coeﬃcientwise c (n) ≥ cS (n). Therefore the n-th coeﬃcient in exp(H (s)) is not smaller than the n-th coefﬁcient, say c (n), in exp(CS (s)). Hence CM (S) (n) ≥ c (n). Let U (u) := i≤u c (i). We now have by (5) ∞ s· e −su U (u)du =

∞ n=0

0

c (n)e −sn

1 = exp (1 + o(1))exp((1 − ))κ d . s

248

ANDREAS WEIERMANN

Then we obtain by the Tauberian theorem of Wagner [34] that u log(U (u)) ∼ (1 − )κ · 1 , (log(u)) d and hence the assertion, if a corresponding Tauberian condition is satisﬁed. This is indeed so since for g(s) = exp((1 + )κ s1d ) we have 0 < g(s) → ∞ and for some ﬁxed (g (s))2 ≤1 0 0. Let 1d d expk−1 ((1 − )κt) s dt f(s) = t d +1 1 and κ(1 − ) . h(v) = d logk−1 (v) (Note that again this term is exactly of the form of equation (21.7) in [22].) Since by induction hypothesis

u U (u) ≥ expk−1 (1 − o(1))κ logk−3 (u) we can choose a subset of T ⊇ {α < k−1 } such that cTS (≤ v) ∼

exp(vh(v) + f(h(v))) . h(v) 2f (h(v))

Then Laplace’s method yields

∞ ∞ 1 −su −sn (6) e cS (≤ u)du = cS (n)e ∼ expk−1 (1 − )κ d s· s 0 n=0

as s → 0.

ANALYTIC COMBINATORICS OF THE TRANSFINITE

249

Let M (S) be the set of ﬁnite (normal form) sums (multisets) of elements from S. Then M (S) is contained in {α : α < k (d )} so that ck (d ) (n) ≥ cM (S) (n). So it suﬃces to get a good lower bound on cM (S) (n). As before ∞ ∞ let CS (s) := n=0 cS (n)e −sn , let H (s) := m=0 m1 CS (sm) and assume that ∞ H (s) = n=0 c (n)e −sn . Then coeﬃcientwise c (n) ≥ cS (n). Therefore the n-th coeﬃcient in exp(H (s)) is not smaller than the n-th coeﬃcient, say c (n), in exp(CS (s)). Hence CM (S) (n) ≥ c (n). Let U (u) := i≤u c (i). We now have by (6) ∞ s· e −su cM (S) (≤ u)du =

∞ n=0

= exp

0

cM (S) (n)e −sn = exp(H (s)) ≥ exp(CS (s)) ∞

cS (n)e

−sn

=

n=0

∞

c (n)e −sn

n=0

1 . = exp((1 + o(1))) · expk−1 κ (1 − ) d s The Tauberian theorem of Wagner [34] yields log(U (u)) ∼ (1 − )κ · u 1 , hence the assertion, if a corresponding Tauberian condition is (log(k−2) (u)) d satisﬁed. This is indeed true since for g(s) = expk−1 (1 − )κ s1d we have 0 < g(s) → ∞ and for some ﬁxed 0 αn then let MAl α := 2MAl −1 α1 + · · · + 2MAl −1 αn . We guess that the asymptotic behavious of the resulting count function behaves similarly as the asymptotic of c Al .

ANALYTIC COMBINATORICS OF THE TRANSFINITE

259

Still another coding of ordinals less than l +1 for ﬁxed l has roots going back to Hardy and Ramanujan [18]. Deﬁnition 7. 1. If α = k1 + · · · + kn < with k1 ≥ · · · ≥ kn then HR1 α := p1k1 +1 · · · · · pnkn +1 . 2. For α = α1 + · · · + αn < l +1 (where l > 1) with α1 ≥ · · · ≥ αn put HR α HR α HRl α := p1 l −1 1 ·· · ··pn l −1 n if α = α1 +· · ·+ αn ≥ α1 ≥ · · · ≥ αn . For ≤ l +1 put

cHRl (n) := α < : HRl α ≤ n . log(n) d 1 1 Theorem 5. 1. cHR . d (n) ∼ d !d ! log(log(n)) log(n) 2 1 2. log(cHR . (n)) ∼ √ log(log(n)) 3 Proof. The ﬁrst assertion follows e.g. from Karamata’s theorem as shown in [36]. The ﬁrst assertion can also be proved by a more elementary argument of Schlage Puchta (private communication). The second assertion was proved by Hardy and Ramanujan [18].

We expect that Parameswaran’s theorem provides weak asympotics for the count functions related to higher -powers. Finally we would like to mention a (perhaps somewhat artiﬁcial) norm function which leads to fractional exponents for partitions. Deﬁnition 8. 1. Nk 0 := 0 2. Nk α := (1 + Nk α1 )k + · · · + (1 + Nk αn )k if α = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn . We guess that the asymptotic for the resulting count functions can be determined using Petrogradsky’s theorem starting at its base with d

cNdk (≤ n) ∼ constant · n k . The corresponding multiplicative analogue will be given by Deﬁnition 9. 1. Sk 0 := 1 2. Sk α := pSkk α1 · · · · · pSkk αn if α = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn . We expect that the asymptotic will start with something like d

cSkd (≤ n) ∼ constant · (log n) k . The choice of Nk and Sk is of course somewhat artiﬁcial. But, and this seems to be of certain interest, if one uses Nk in place of N and Sk in place of S in Theorems 7 and 8 of the next section it will turn out that the threshold function will not change. So there will be a phenomenon of structural stability concerning applications of diﬀerent norms.

260

ANDREAS WEIERMANN

§4. Applications to phase transition results for G¨odel incompleteness. We start with some standard conventions. As usual the least integer greater than or equal to a given real number x is denoted by x and similarly the greatest integer smaller than or equal to a given real number x is denoted by x. The binary length |n| of a natural number n is deﬁned by |n| := log(n + 1). The d -times iterated length function |·|d is deﬁned recursively as follows |x|0 := x and |x|d +1 := ||x|d |. (This function mainly serves as an LPA deﬁnable version of log(d ) .) As usual we assume that the ordinals less than ε0 are available in PA via a standard coding. Recall that for α < ε0 the number Nα denotes the number of occurrences of in the Cantor normal form of α. Let CWO(N, α, f) be the following principle ∀K ∃M B(N, α, f, K, M ) where B(N, α, f, K, M ) is the assertion ! ! " (∀α0 , . . . , αM < α) (∀i ≤ M ) Nαi ≤ K + f(i) =⇒ ! "" (∃i, j < ) 0 ≤ i < j ≤ M & αi ≤ αj . Theorem 6 (Friedman). Let d ≥ 0 be a ﬁxed natural number. Then IΣd +1 CWO(N, d +2 , f) where f is the identity map. Proof. See, for example, [33] for a proof.

Before we can sharpen this result we need to recall the deﬁnition of the standard system of canonical fundamental sequences and the Hardy hierarchy. Deﬁnition 10. 1. If = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn and αn = αn + 1 then [x] := α1 + · · · + αn · x. 2. If = α1 + · · · + αn ≥ α1 ≥ · · · ≥ αn and αn is a limit then [x] := α1 + · · · + αn [x] . Deﬁnition 11 (The Hardy hierarchy). H0 (x) := x,

Hα+1 (x) := Hα (x + 1),

H (x) := H [x] (x) if is a limit.

The basic theory of the Hardy hierarchy is developed, for example, in [7] or [31]. The most fundamental result concerning this hierarchy is that (Hα )α α1 > · · · > αM with Nαi ≤ K + i and α0 < d +2 . Our aim is to transform this sequence into a strictly descending sequence −1 0 > 1 > · · · > M with N i ≤ p(K ) + |i| · Hd +2 (i) |i|d for i ≤ M where p is a primitive recursive function depending only on K . If this is achieved we are done since M depends on K in a way which is not provably recursive in IΣd +1 . From Nα0 ≤ K we conclude α0 < d +1 (K − d − 1). For the sequel assume that D depending primitive recursively on K is large enough so that the asymptotic bounds apply. For i ≥ D put Mi := α < d +1 (K + 3) : Nα ≤ |i| · K+1 |i|d and let enumi (l ) be the l -th element of Mi . Then we deﬁne i := d +1 (K + 3) · α|i| + enumi 2|i| − i .

262

ANDREAS WEIERMANN

(This transformation might be interpreted as a renormalization operator on descending sequences.) The sequence i is strictly decreasing. Moreover we ﬁnd for i ≥ D Ni ≤ (d + 1 + K + 3) · (K + |i|) + |i| · K+1 |i|d ≤ K + |i| · K |i|d . Now we sharpen this slightly using a trick due to Arai [4]. For i ≤ F (K ) we (i) ≤ F −1 (i) ≤ F −1 (F (K )) = K . Therefore have H−1 d +2 −1 Ni ≤ K + |i| · Hd +2 (i) |i|d holds for i ≥ D. Now we have to check the side conditions. For i < D we simply put i := d +1 (K + 3) · d +1 (K + 3) + D − i. We are nearly through but one last point has to be checked namely the well-deﬁnedness of i . For this it suﬃces to check that Mi has at least 2|i| many elements. This is the place where we need the bounds from analytic combinatorics. Case 1. d = 0. We have by assertion 1 of Corollary 1 and obvious asymptotic calculations for a suitable and constant C0,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable constant D0,K K+2 |Mi | ≥ C0,K · |i| · K+1 |i|d √ K+2 K+1 ≥ C0,K · i K+2

≥ C0,K · i K+1 ≥ 2|i| . Case 2. d = 1. We have by assertion 2 of Corollary 1 and obvious asymptotic calculations for a suitable and a constant C1,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable constant D1,K √ K+3 K+4 C1,K |i|·

|Mi | ≥ 2

K+3

K+4 C1,K |i|

≥2

K+3 K+4

≥ 2C1,K ≥ 2|i| .

K+1

1+

|i|

1 K+1

K+2 · K+3 K+4

|i| K+1

K+3 K+4

ANALYTIC COMBINATORICS OF THE TRANSFINITE

263

Case 3. d ≥ 2. We have by assertions 3 and 4 of Corollary 1 and obvious asymptotic calculations for a suitable and a constant Cd,K (which is primitive recursively computable with respect to K ) and for all i larger than a suitable and primitive recursively computable (with respect to K ) constant Dd,K √ |i|· K+1 |i|d √ Cd,K √ K+3 ||i|· K+1 |i| | d d −1 |Mi | ≥ 2 ≥ 2|i| since K+1 |i|d ||i| · K+1 |i|d |d −1 → 0 as i → ∞. Now we prove the second assertion. Fix < d +2 . Given K let M := 2H (K·2)+1 . This is well deﬁned in IΣd +1 . Moreover we may assume that M is large enough for asymptotic purposes. We claim that this M is suﬃcient for our purposes. Assume otherwise that α0 > α1 > · · · > αM is a sequence such −1 that Nαi ≤ K + |i| · H (i) |i|d and α0 < d +2 . Then α0 < d +1 (K − d − 1). We have for i ≥ M2 that H −1 (i) ≥ H −1 (H (2K )) = 2K and hence for i ≥ n2 αi ∈ α < d +1 (K ) : Nα ≤ |M | 2K |M |d . # K+3

The number of elements of the latter set is at least M2 as witnessed by the αi . We obtain the contradiction by calculating bounds on the number of elements on this set. To this end we distinguish three cases. Case 1. d = 0. By choice of M we may assume that the bounds of assertion 1 of Corollary 1 apply and that for a certain constant C0,K we have √ K M 2K < C0,K · |M | M 2 √ = C0,K · (|M |)K M M . < 2 Case 2. d = 1. By choice of M we may assume that the bounds from assertion 2 of Corollary 1 apply and that for a certain constant C1,K we have K √ K+1 2K M < 2C1,K · |M | |M | 2 = 2C1,K ·(|M |) M . < 2

2K+1 K 2K K+1

264

ANDREAS WEIERMANN

Case 3. d ≥ 2. By choice of M we may assume that the bounds from assertions 3 and 4 of Corollary 1 apply and that for a certain constant Cd,K we have √ Cd,K · |M | 2K |M |d √ K M |M | 2K |M |d −1 d −1

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close