The Ultrapower Axiom 9783110719734, 9783110719697

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Gabriel Goldberg The Ultrapower Axiom

De Gruyter Series in Logic and Its Applications

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Edited by Itaï Ben Yaacov, Université Claude Bernard, France Denis R. Hirschfeldt, University of Chicago, USA Ieke Moerdijk, Utrecht University, The Netherlands Itay Neeman, UCLA, USA Menachem Magidor, The Hebrew University of Jerusalem, Israel Anand Pillay, University of Leeds, United Kingdom

Volume 10

Gabriel Goldberg

The Ultrapower Axiom |

Mathematics Subject Classification 2020 03E05, 03E45, 03E55, 03E65 Author Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 USA [email protected]

ISBN 978-3-11-071969-7 e-ISBN (PDF) 978-3-11-071973-4 e-ISBN (EPUB) 978-3-11-071979-6 ISSN 1438-1893 Library of Congress Control Number: 2022932968 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Acknowledgments Most of the results contained in this book were proved while I was a graduate student under Hugh Woodin from 2015 to 2019. His work raised many of the central questions addressed in this monograph, and his intuition never failed to set me on the path to the answers. I also benefited from many conversations and university-sponsored lunches with my unofficial adviser, Peter Koellner, at whose suggestion I wrote this book. In March 2017, John Steel hosted me for a very productive month at UC Berkeley, during which many of the main ideas behind this work began to emerge. I want to thank him for the many discussions we had at the time as well as during my postdoc at Berkeley over the past 2 years. I am sincerely grateful to everyone who has offered comments on parts of this book or engaged with the research it contains. In particular, I thank Arthur Apter, Moti Gitik, Akihiro Kanamori, Grigor Sargsyan, Trevor Wilson and the anonymous reviewers. I also thank Nadja Schedensack for her patience throughout the editorial process. I owe a great debt to my peers, with inspired me to learn logic and influenced my work more than I care to admit. I especially thank Juan Aguilera, Alexander Bertoloni Meli, Doug Blue, Justin Cavitt, Elliot Glazer, Patrick Lutz, Benny Siskind, and Jing Zhang. Finally, I want to thank the many people outside of mathematics who supported me throughout the research and writing of this book despite my inability to explain even vaguely what it is about, particularly my parents, my brothers, and my friends Grant, Jack, Nick, Stefan, and Tim. Most of all, thank you, TJ, for believing in me when I couldn’t. You mean so much to me, and you always will.

https://doi.org/10.1515/9783110719734-201

Contents Acknowledgments | V 1 1.1

Introduction | 1 Outline | 2

2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5

The linearity of the Mitchell order | 7 Introduction | 7 Normal ultrafilters and the Mitchell order | 7 The ultrapower axiom | 10 Outline of Chapter 2 | 10 Preliminary definitions | 11 Inner models | 11 Ultrapowers | 12 Close embeddings | 16 Uniform ultrafilters | 18 The Mitchell order on normal ultrafilters | 20 The linearity of the Mitchell order | 20 Weak comparison | 21 Weak comparison and the Mitchell order | 23 Weak comparison and the ultrapower axiom | 26 The ultrapower axiom and the Mitchell order | 29 Technical lemmas related to weak comparison | 31

3 3.1 3.1.1 3.1.2 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.2 3.3.3 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.5

The Ketonen order | 35 Introduction | 35 Ketonen’s order | 35 Outline of Chapter 3 | 35 Preliminary definitions | 37 Fine ultrafilters on ordinals | 37 Limits of ultrafilters | 39 The Ketonen order | 41 Characterizations of the Ketonen order | 41 Basic properties of the Ketonen order | 44 The global Ketonen order | 48 Orders on ultrafilters | 50 The Mitchell order | 50 The Rudin–Keisler order | 51 The Lipschitz order | 56 The Ketonen order on filters | 60 The linearity of the Ketonen order | 63

VIII | Contents 4 4.1 4.1.1 4.1.2 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.2.5 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4

The generalized Mitchell order | 71 Introduction | 71 The linearity of the generalized Mitchell order | 71 Outline of Chapter 4 | 71 Folklore of the generalized Mitchell order | 72 Strength and the Mitchell order | 72 Supercompactness and the Mitchell order | 76 The Kunen inconsistency | 81 The well foundedness of the generalized Mitchell order | 83 The nonlinearity of the generalized Mitchell order | 85 Dodd soundness | 87 Introduction | 87 Dodd sound embeddings, extenders, and ultrafilters | 87 The generalized Mitchell order on Dodd sound ultrafilters | 95 Generalizations of normality | 98 Normal fine ultrafilters | 100 Weakly normal ultrafilters | 102 Solovay’s lemma | 106 Supercompactness and singular cardinals | 109

5 5.1 5.1.1 5.1.2 5.2 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.5.1 5.5.2 5.5.3 5.5.4

The Rudin–Frolík order | 117 Introduction | 117 Ultrafilters on the least measurable cardinal | 117 Outline of Chapter 5 | 118 The Rudin–Frolík order | 119 Below the first μ-measurable cardinal | 125 Introduction | 125 Irreducible ultrafilters and μ-measurability | 126 Factorization into irreducibles | 129 The structure of the Rudin–Frolík order | 135 The local ascending chain condition | 135 Pushouts and the Rudin–Frolík lattice | 136 The finiteness of the Rudin–Frolík order | 142 Translations and limits | 147 The internal relation | 150 A generalized Mitchell order | 150 The Mitchell order versus the internal relation | 151 Basic theory of the internal relation | 153 Commuting ultrapowers and wellfoundedness | 157

6 6.1

V = HOD and GCH from UA | 163 Introduction | 163

Contents | IX

6.1.1 6.1.2 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7 6.3.8

The universe above a supercompact cardinal | 163 Outline of Chapter 6 | 163 Ordinal definability | 164 The generalized continuum hypothesis | 169 Introduction | 169 The number of supercompactness measures | 169 The local capturing property | 171 λ-Mitchell ultrafilters | 172 λ-Mitchell ultrafilters from UA | 174 GCH from UA | 177 ♦ on the critical cofinality | 178 The size and saturation of the Vopěnka algebra | 179

7 7.1 7.1.1 7.1.2 7.2 7.2.1 7.2.2 7.2.3 7.3 7.3.1 7.3.2 7.3.3 7.3.4 7.3.5 7.3.6 7.4 7.4.1 7.4.2 7.4.3 7.4.4 7.5 7.5.1 7.5.2 7.5.3 7.5.4

The least supercompact cardinal | 183 Introduction | 183 The identity crisis | 183 Outline of Chapter 7 | 184 Strong compactness | 185 Some characterizations of strong compactness | 185 Ketonen’s theorem | 188 Solovay’s theorem | 190 The least ultrafilter Kλ | 192 Fréchet cardinals | 192 Ketonen ultrafilters | 194 Introducing Kλ | 196 The universal property of Kλ | 197 Independent families and the Hamkins properties | 200 The strength and supercompactness of Kλ | 205 Fréchet cardinals | 208 The next Fréchet cardinal | 208 The strong compactness of κλ | 211 The least supercompact cardinal | 215 The number of countably complete ultrafilters | 217 Isolation | 220 Isolated measurable cardinals | 220 Ultrafilters on an isolated cardinal | 223 Isolated cardinals and the GCH | 232 The linearity of the Mitchell order without GCH | 242

8 8.1 8.1.1

Higher supercompactness | 249 Introduction | 249 Obstructions to the supercompactness analysis | 249

X | Contents 8.1.2 8.1.3 8.1.4 8.1.5 8.2 8.2.1 8.2.2 8.2.3 8.2.4 8.3 8.3.1 8.3.2 8.3.3 8.4 8.4.1 8.4.2 8.4.3 9 9.2.2 9.2.3 9.3.1 9.3.2 9.4.1 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.6.1 9.6.2 9.7.1 9.7.2 9.7.3

Menas’ theorem | 249 Complete UA | 251 Irreducible ultrafilters and supercompactness | 252 Outline of Chapter 8 | 254 The irreducibility theorem | 255 Tightness and irreducibility | 255 Translations of Kλ | 257 Elementary embeddings and normal filters | 259 A proof of the irreducibility theorem | 262 Resolving the identity crisis | 269 The equivalence of strong compactness and supercompactness | 270 Level-by-level equivalence at singular cardinals | 275 The Mitchell order, the internal relation and coherence | 280 Very large cardinals | 289 Huge cardinals | 289 Cardinal preserving elementary embeddings | 293 Supercompactness at inaccessible cardinals | 298 Open questions | 307 Weak comparison | 307 The ultrapower axiom | 307 The Ketonen order | 308 The Lipschitz order | 309 The generalized Mitchell order | 310 The Rudin–Frolík order | 311 The internal relation | 312 V = HOD | 312 The generalized continuum hypothesis | 313 The ground axiom | 314 Isolated cardinals | 314 Local supercompactness | 315 The complete ultrapower axiom | 316 Cardinal preserving embeddings | 316 Supercompactness at inaccessible cardinals | 317

Bibliography | 319 Index | 321

1 Introduction Gödel, in his 1947 paper, “What is Cantor’s continuum problem,” was the first to suggest that even those questions that cannot be answered using the commonly accepted ZFC axioms of set theory might be resolved in a principled way by adopting axioms that “assert the existence of still further iterations” of the powerset operation. Though the strong principles central to this monograph are admittedly wild extrapolations of Gödel’s early intuitions, this remains the driving idea behind large cardinal axioms. Such axioms have been remarkably successful in settling classical problems left open by the ZFC axioms, but many problems, including Cantor’s continuum problem, remain unsolvable under any of the known large cardinal hypotheses. Results of Lévy– Solovay [27] and others suggest that these problems cannot be solved using any large cardinal hypothesis that will ever be formulated. Are there further principles which in conjunction with large cardinal axioms resolve all set theoretic questions? To answer this question, set theorists have sought to construct canonical models of set theory, free from the ambiguity inherent in the concept of set. The simplest example of such a model is Gödel’s constructible universe L, the smallest model of Zermelo–Fraenkel set theory that contains every ordinal number. One sense in which L is canonical is that seemingly every question about its internal structure can be answered. For example, Gödel proved that L satisfies the continuum hypothesis. In contrast, many of the most basic properties of the universe of all sets V, the maximum model of set theory, cannot be determined in any commonly accepted axiomatic system. To what extent does L provide a good approximation to the universe of sets? On the one hand, the principle that every set belongs to L (or in other words, V = L) cannot be refuted using the ZFC axioms, since L itself is a model of the theory ZFC + V = L. If V = L, then L approximates V very well. On the other hand, the model L fails to satisfy relatively weak large cardinal axioms. If one takes the stance that these large cardinal axioms are true in the universe of sets, one must conclude that V ≠ L. Moreover, it follows from large cardinal axioms that L constitutes only a tiny fragment of the universe of sets. For example, assuming large cardinal axioms, the set of real numbers that lie in L is countable. Are there canonical models generalizing L that yield better approximations to V? A whole subfield of set theory known as inner model theory is devoted to answering this question. It turns out that there is a hierarchy of canonical models beyond L, satisfying stronger and stronger large cardinal axioms. The program of building such models has met striking success, reaching large cardinal axioms as strong as a Woodin limit of Woodin cardinals. Based on the pattern that has emerged so far, it seems plausible that every large cardinal axiom has a corresponding canonical model. At present, however, a vast expanse of large cardinal axioms are not known to admit canonical models. A key target problem for inner model theory is the construction of a canonical model with a supercompact cardinal. Work of Woodin suggests that the https://doi.org/10.1515/9783110719734-001

2 | 1 Introduction solution to this problem alone will yield an ultimate canonical model that inherits essentially all large cardinals present in the universe. There is therefore hope that the goal of constructing inner models for all large cardinal axioms might be achieved in a single stroke. If this is possible, the resulting model would be of enormous set theoretic interest, since it would closely approximate the universe of sets and yet admit an analysis that is as detailed as that of Gödel’s L. This monograph investigates whether there can be a canonical model with a supercompact cardinal. To do this, we develop an abstract approach to inner model theory. This is accomplished by introducing a combinatorial principle called the ultrapower axiom (UA), which is expected to hold in all canonical models. If one could show that UA is inconsistent with a supercompact cardinal, one would arguably have to conclude that there can be no canonical model with a supercompact cardinal. Supplemented with large cardinal axioms, UA turns out to have surprisingly strong and coherent consequences for the structure of the upper reaches of the universe of sets, particularly above the first supercompact cardinal. These consequences are entirely consistent with what one would expect to hold in a canonical model, yet are proved by methods that are completely different from the usual techniques of inner model theory. The coherence of this theory provides compelling evidence that UA is consistent with a supercompact cardinal. If this is the case, it seems that the only possible explanation is that the canonical model for a supercompact cardinal does indeed exist. Optimistically, studying the consequences of UA will shed light on how this model should be constructed.

1.1 Outline We now describe the main results of this monograph. Chapter 2 In this introductory chapter, we introduce UA in the context of the problem of the linearity of the Mitchell order on normal ultrafilters. We show first that UA holds in all canonical inner models, a result that is philosophically central to this monograph. More precisely, we prove that UA is a consequence of Woodin’s weak comparison principle. Theorem 2.3.10. Assume that V = HOD and there is a Σ2 -correct worldly cardinal. If weak comparison holds, then UA holds. We then show that UA implies the linearity of the Mitchell order. Theorem 2.3.11 (UA). The Mitchell order is linear.

1.1 Outline

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Two applications of this result to longstanding problems of Solovay–Reinhardt– Kanamori [41] are explained in the introduction to Chapter 2. Chapter 3 This chapter introduces the Ketonen order, a generalization of the Mitchell order to all countably complete ultrafilters on ordinals. The restriction of this order to weakly normal ultrafilters was originally introduced by Ketonen. The first proof of the wellfoundedness of the more general order is due to the author. Theorem 3.3.8. The Ketonen order is well-founded. The main theorem of this chapter explains the fundamental role of the Ketonen order in applications of UA. Theorem 3.5.1. The linearity of the Ketonen order is equivalent to UA. In addition, we explain the relationship between the Ketonen order and various well-known orders like the Rudin–Keisler order and the Mitchell order. Chapter 4 The topic of this chapter is the generalized Mitchell order, which is defined in exactly the same way as the usual Mitchell order on normal ultrafilters but removing the requirement that the ultrafilters involved be normal. This order is not linear (assuming there is a measurable cardinal), and in fact it is quite pathological when considered on ultrafilters in general. The two main results of this chapter generalize the linearity of the Mitchell order to nice classes of ultrafilters. Theorem 4.3.29 (UA). The generalized Mitchell order is linear on Dodd sound ultrafilters. Dodd soundness is a generalization of normality that was first isolated in the context of inner model theory by Steel [36]. A uniform ultrafilter U on a cardinal λ is Dodd sound if the map h : P(λ) → MU defined by h(X) = jU (X) ∩ [id]U and belongs to MU . The concept is discussed at great length in Section 4.3. A better known generalization of normality is the concept of a normal fine ultrafilter (Definition 4.4.7), introduced by Solovay, and underpinning the theory of supercompact cardinals. The second result of this chapter generalizes the linearity of the Mitchell order to this class of ultrafilters. Theorem 4.4.2 (UA). Suppose λ is a cardinal such that 2