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Geometric Set Theory Mathematical Surveys and Monographs Volume 248 Geometric Set Theory Paul B. Larson Jindˇrich Zapletal 10.1090/surv/248 Geometric Set Theory Mathematical Surveys and Monographs Volume 248 Geometric Set Theory Paul B. Larson Jindˇrich Zapletal EDITORIAL COMMITTEE Robert Guralnick, Chair Natasa Sesum Bryna Kra Constantin Teleman Melanie Matchett Wood 2010 Mathematics Subject Classification. Primary 03E15, 03E25, 03E35, 03E40, 05C15, 05B35, 11J72, 11J81, 37A20. For additional information and updates on this book, visit www.ams.org/bookpages/surv-248 Library of Congress Cataloging-in-Publication Data Names: Larson, Paul B. (Paul Bradley), 1970– author. | Zapletal, Jindˇrich, 1969– author. Title: Geometric set theory / Paul B. Larson, Jindrich Zapletal. Description: Providence, Rhode Island: American Mathematical Society, [2020] | Series: Mathe- matical surveys and monographs, 0076-5376; volume 248 | Includes bibliographical references and index. Identifiers: LCCN 2020009795 | ISBN 9781470454623 (softcover) | ISBN 9781470460181 (ebook) Subjects: LCSH: Descriptive set theory. | Equivalence relations (Set theory) | Borel sets. | Independence (Mathematics) | Axiomatic set theory. | Forcing (Model theory) | AMS: Mathe- matical logic and foundations – Set theory – Descriptive set theory. | Mathematical logic and foundations – Set theory – Axiom of choice and related propositions. | Mathematical logic and foundations – Set theory – Consistency and independence results. | Mathematical logic and foundations – Set theory – Other aspects of forcing and Boolean-valued models. | Com- binatorics – Graph theory – Coloring of graphs and hypergraphs. | Combinatorics – Designs and configurations – Matroids, geometric lattices. | Number theory – Diophantine approxima- tion, transcendental number theory– Irrationality; linear independence over a field. | Number theory – Diophantine approximation, transcendental number theory – Transcendence (gen- eral theory). | Dynamical systems and ergodic theory – Ergodic theory – Orbit equivalence, cocycles, ergodic equivalence relations. Classification: LCC QA248 .L265 2020 | DDC 511.3/22–dc23 LC record available at https://lccn.loc.gov/2020009795 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2020 by the authors. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 Contents Preface ix Chapter 1. Introduction 1 1.1. Outline of the subject 1 1.2. Equivalence relation results 2 1.3. Independence: by topic 5 1.4. Independence: by model 10 1.5. Independence: by preservation theorem 12 1.6. Navigation 15 1.7. Notation and terminology 25 Part I. Equivalence relations 33 Chapter 2. The virtual realm 35 2.1. Virtual equivalence classes 35 2.2. Virtual structures 37 2.3. Classification: general theorems 39 2.4. Classification: specific examples 41 2.5. Cardinal invariants 45 2.6. Restrictions on partial orders 58 2.7. Absoluteness 61 2.8. Dichotomies 64 Chapter 3. Turbulence 73 3.1. Independent functions 73 3.2. Examples and operations 75 3.3. Placid equivalence relations 78 3.4. Examples and operations 79 3.5. Absoluteness 83 3.6. A variation for measure 86 Chapter 4. Nested sequences of models 93 4.1. Prologue 93 4.2. Coherent sequences of models 93 4.3. Choice-coherent sequences of models 96 v vi CONTENTS Part II. Balanced extensions of the Solovay model 103 Chapter 5. Balanced Suslin forcing 105 5.1. Virtual conditions 105 5.2. Balanced virtual conditions 108 5.3. Weakly balanced Suslin forcing 114 Chapter 6. Simplicial complex forcings 117 6.1. Basic concepts 117 6.2. Fragmented complexes 117 6.3. Matroids 124 6.4. Quotient variations 128 Chapter 7. Ultrafilter forcings 135 7.1. A Ramsey ultrafilter 135 7.2. Fubini powers of the Fréchet ideal 136 7.3. Ramsey sequences of structures 138 7.4. Semigroup ultrafilters 141 Chapter 8. Other forcings 145 8.1. Coloring graphs 145 8.2. Coloring hypergraphs 148 8.3. Discontinuous homomorphisms 157 8.4. Automorphisms of 풫(휔) modulo finite 159 8.5. Kurepa families 160 8.6. Set mappings 162 8.7. Saturated models on quotient spaces 165 8.8. Non-DC variations 169 8.9. Side condition forcings 170 8.10. Weakly balanced variations 173 Chapter 9. Preserving cardinalities 179 9.1. The well-ordered divide 179 9.2. The smooth divide 182 9.3. The turbulent divide 188 9.4. The orbit divide 193 9.5. The 피퐾휍 divide 205 9.6. The pinned divide 211 Chapter 10. Uniformization 213 10.1. Tethered Suslin forcing 213 10.2. Uniformization theorems 214 10.3. Examples 221 Chapter 11. Locally countable structures 227 11.1. Central objects and notions 227 11.2. Definable control 233 11.3. Centered control 237 11.4. Linked control 244 CONTENTS vii 11.5. Measured control 250 11.6. Ramsey control 254 11.7. Liminf control 258 11.8. Compactly balanced posets 262 Chapter 12. The Silver divide 269 12.1. Perfectly balanced forcing 269 12.2. Bernstein balanced forcing 274 12.3. 푛-Bernstein balanced forcing 285 12.4. Existence of generic filters 292 Chapter 13. The arity divide 299 13.1. 푚, 푛-centered and balanced forcings 299 13.2. Preservation theorems 300 13.3. Examples 307 Chapter 14. Other combinatorics 313 14.1. Maximal almost disjoint families 313 14.2. Unbounded linear suborders 314 14.3. Measure and category 315 14.4. The Ramsey ultrafilter extension 318 Bibliography 323 Index 329 Preface We wish to present to the reader a fresh and exciting new area of mathematics: geometric set theory. The purpose of this research direction is to compare transitive models of set theory with respect to their extensional agreement and definability. It turns out that many fracture lines in descriptive set theory, analysis, and model theory can be efficiently isolated and treated from this point of view. A particular success is 2 the comparison of various Σ1 consequences of the Axiom of Choice in unparalleled detail and depth. The subject matter of the book was rather slow in coming. The initial work, restat- ing Hjorth’s turbulence in geometric terms and isolating the notion of a virtual quotient space of an analytic equivalence relation, existed in rudimentary versions since about 2013 in unpublished manuscripts of the second author. The joint effort74 [ ] contained some independence results in choiceless set theory similar to those of the present book, but in a decidedly suboptimal framework. It was not until the January 2018 discovery of balanced Suslin forcing that the flexibility and power of the geometric method fully asserted itself. The period after that discovery was filled with intense wonder—passing from one configuration of models of set theory to another and testing how theysepa- rate various well-known concepts in descriptive set theory, analysis, and model theory. At the time of writing, geometric set theory seems to be an area wide open for innu- merable applications. The authors benefited from discussion with a number of mathematicians, includ- ing, but not limited to, David Chodounský, James Freitag, Michael Hrušák, Anush Tserunyan, Douglas Ulrich, and Lou van den Dries. The second author wishes to ex- tend particular thanks to the Bernoulli Center at EPFL Lausanne, where a significant part of the results was obtained during the special semester on descriptive set theory and Polish groups in 2018. During the work on the book, the first author was supported by grants NSF DMS 1201494 and DMS-1764320. The second author was supported by NSF grant DMS 1161078. ix Bibliography [1] Francis Adams and Jindřich Zapletal, Cardinal invariants of closed graphs, Israel J. Math. 227 (2018), no. 2, 861–888, DOI 10.1007/s11856-018-1745-6. MR3846345 [2] Martin Aigner, Combinatorial theory, Grundlehren der Mathematischen Wissenschaften [Fundamen- tal Principles of Mathematical Sciences], vol. 234, Springer-Verlag, Berlin-New York, 1979. MR542445 [3] Reinhold Baer, Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc. 46 (1940), 800–806, DOI 10.1090/S0002-9904-1940-07306-9. MR2886 [4] Bohuslav Balcar, Thomas Jech, and Jindřich Zapletal, Semi-Cohen Boolean algebras, Ann. Pure Appl. Logic 87 (1997), no. 3, 187–208, DOI 10.1016/S0168-0072(97)00009-2. MR1474561 [5] John T. Baldwin and Paul B. Larson, Iterated elementary embeddings and the model theory of infinitary logic, Ann. Pure Appl. Logic 167 (2016), no. 3, 309–334, DOI 10.1016/j.apal.2015.12.004. MR3437649 [6] John T. Baldwin, Sy D. Friedman, Martin Koerwien, and Michael C. Laskowski, Three red her- rings around Vaught’s conjecture, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3673–3694, DOI 10.1090/tran/6572. MR3451890 [7] Tomek Bartoszyński and Haim Judah, Set theory: On the structure of the real line, A K Peters, Ltd., Wellesley, MA, 1995. MR1350295 [8] James E. Baumgartner and Alan D. Taylor, Partition theorems and ultrafilters, Trans. Amer. Math. Soc. 241 (1978), 283–309, DOI 10.2307/1998845. MR491193 [9] A. Bella, A. Dow, K. P. Hart, M. Hrušák, J. van Mill, and P. Ursino, Embeddings into 풫(ℕ)/fin and exten- sion of automorphisms, Fund. Math. 174 (2002), no. 3, 271–284, DOI 10.4064/fm174-3-7. MR1925004 [10] Mariam Beriashvili, Ralf Schindler, Liuzhen Wu, and Liang Yu, Hamel bases and well-ordering the con- tinuum, Proc.
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