Deconstructing Inner Model Theory
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Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
Are Large Cardinal Axioms Restrictive?
Are Large Cardinal Axioms Restrictive? Neil Barton∗ 24 June 2020y Abstract The independence phenomenon in set theory, while perva- sive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality prin- ciples depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardi- nals are still important axioms of set theory and can play many of their usual foundational roles. Introduction Large cardinal axioms are widely viewed as some of the best candi- dates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical con- sequences for questions independent from ZFC (such as consistency statements and Projective Determinacy1), and appear natural to the ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@uni- konstanz.de. yI would like to thank David Aspero,´ David Fernandez-Bret´ on,´ Monroe Eskew, Sy-David Friedman, Victoria Gitman, Luca Incurvati, Michael Potter, Chris Scam- bler, Giorgio Venturi, Matteo Viale, Kameryn Williams and audiences in Cambridge, New York, Konstanz, and Sao˜ Paulo for helpful discussion. Two anonymous ref- erees also provided helpful comments, and I am grateful for their input. I am also very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foundations of Mathematics. -
UC Irvine UC Irvine Electronic Theses and Dissertations
UC Irvine UC Irvine Electronic Theses and Dissertations Title Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms Permalink https://escholarship.org/uc/item/9875g511 Author Schatz, Jeffrey Robert Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Philosophy by Jeffrey Robert Schatz Dissertation Committee: Distinguished Professor Penelope Maddy, Chair Professor Toby Meadows Dean’s Professor Kai Wehmeier Professor Jeremy Heis 2019 All materials c 2019 Jeffrey Robert Schatz DEDICATION To Mike and Lori Schatz for always encouraging me to pursue my goals AND In memory of Jackie, a true and honest friend. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iv CURRICULUM VITAE vi ABSTRACT OF THE DISSERTATION vii CHAPTER 1: Two Contemporary Candidates for A Strong Theory of Sets 1 1 The Axiomatization of Set Theory and The Question of Axiom Selection 2 1.1 The Inner Model Program 14 1.2 The Forcing Axiom Program 27 1.3 How to Settle This Dispute? 36 1.4 CHAPTER 2: Axiom Selection and the Maxim of ‘Maximize’ 38 2 ‘Maximize’ as a Methodological Maxim 39 2.1 Two Notions of Maximize 43 2.2 Re-characterizing M-Max for Extensions of ZFC 50 2.3 CHAPTER 3: Applying ‘Maximize’ to Contemporary Axiom Candidates 53 3 Towards a Theory for Ultimate L 54 3.1 S-Max: -
Inner Model Theoretic Geology∗
Inner model theoretic geology∗ Gunter Fuchsy Ralf Schindler November 4, 2015 Abstract One of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P -constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ! (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the union of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model. 1 Introduction In [3], the authors introduced several types of inner models which are defined following the paradigm of \undoing" forcing. Thus, the mantle M of a model of set theory V is the intersection of all of its ground models (i.e., the intersection of all ∗AMS MSC 2010: 03E35, 03E40, 03E45, 03E47, 03E55. -
Inner Models from Extended Logics: Part 1*
Inner Models from Extended Logics: Part 1* Juliette Kennedy† Menachem Magidor‡ Helsinki Jerusalem Jouko Va¨an¨ anen¨ § Helsinki and Amsterdam December 15, 2020 Abstract If we replace first order logic by second order logic in the original def- inition of Godel’s¨ inner model L, we obtain HOD ([33]). In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier ([24]), the cofinality quantifier ([35]), or station- ary logic ([6]). Our first set of results show that both L and HOD manifest some amount of formalism freeness in the sense that they are not very sen- sitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals > @1 of V are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. *The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite supported by EPSRC Grant Number EP/K032208/1. The authors are grateful to John Steel, Philip Welch and Hugh Woodin for comments on the results presented here. -
Large Cardinals and the Iterative Conception of Set
Large cardinals and the iterative conception of set Neil Barton∗ 4 December 2017y Abstract The independence phenomenon in set theory, while pervasive, can be par- tially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal ax- ioms consistent with ZFC, since it appears to be ‘possible’ (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several con- ceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. Introduction Large cardinal axioms (discussed in detail below) are widely viewed as some of the best candidates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical consequences for indepen- dence results (both as a tool for generating new models and for consequences within the models in which they reside), and often appear natural to the working set theo- rist, providing fine-grained information about different properties of transfinite sets. They are considered mathematically interesting and central for the study of set the- ory and its philosophy. In this paper, we do not deny any of the above views. We will, however, argue that the status of large cardinal axioms as maximality principles is questionable. -
Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson Doctor of Philosophy in Mathematics University of California, Berkeley Professor John R
Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor John R. Steel, Chair Professor W. Hugh Woodin Professor Sherrilyn Roush Fall 2012 Abstract Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson Doctor of Philosophy in Mathematics University of California, Berkeley Professor John R. Steel, Chair Descriptive inner model theory is the study of connections between descript- ive set theory and inner model theory. Such connections form the basis of the core model induction, which we use to prove relative consistency results relating strong forms of the Axiom of Determinacy with the existence of a strong ideal on }!1 (R) having a certain property related to homogeneity. The main innovation is a unified approach to the \gap in scales" step of the core model induction. 1 Contents Introduction iii Acknowledgements v Chapter 1. Forcing strong ideals from determinacy 1 1.1. The theory \ADR + Θ is regular" 1 1.2. Col(!; R)-generic ultrapowers 2 1.3. A covering property for ideals in generic extensions 5 1.4. The covering property for NS!1;R 8 1.5. A c-dense ideal with the covering property 10 Chapter 2. The core model induction 13 2.1. Model operators 14 2.2. F -mice 18 2.3. The Kc;F construction and the KF existence dichotomy 22 F;] 2.4. M1 from a strong pseudo-homogeneous ideal 28 2.5. -
Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson Doctor of Philosophy in Mathematics University of California, Berkeley Professor John R
Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor John R. Steel, Chair Professor W. Hugh Woodin Professor Sherrilyn Roush Fall 2012 Abstract Contributions to Descriptive Inner Model Theory by Trevor Miles Wilson Doctor of Philosophy in Mathematics University of California, Berkeley Professor John R. Steel, Chair Descriptive inner model theory is the study of connections between descript- ive set theory and inner model theory. Such connections form the basis of the core model induction, which we use to prove relative consistency results relating strong forms of the Axiom of Determinacy with the existence of a strong ideal on }!1 (R) having a certain property related to homogeneity. The main innovation is a unified approach to the \gap in scales" step of the core model induction. 1 Contents Introduction iii Acknowledgements v Chapter 1. Forcing strong ideals from determinacy 1 1.1. The theory \ADR + Θ is regular" 1 1.2. Col(!; R)-generic ultrapowers 2 1.3. A covering property for ideals in generic extensions 5 1.4. The covering property for NS!1;R 8 1.5. A c-dense ideal with the covering property 10 Chapter 2. The core model induction 13 2.1. Model operators 14 2.2. F -mice 18 2.3. The Kc;F construction and the KF existence dichotomy 22 F;] 2.4. M1 from a strong pseudo-homogeneous ideal 28 2.5. -
An Outline of Inner Model Theory
An Outline of Inner Model Theory John R. Steel October 26, 2006 2 Contents I AnOutlineofInnerModelTheory 5 by John R. Steel 1Introduction ............................. 5 2Premice................................ 6 2.1 Extenders .......................... 6 2.2 Fine Extender Sequences . 9 2.3 The Levy Hierarchy, Cores, and Soundness . 13 2.4 Fine structure and ultrapowers . 18 3 IterationTreesandComparison . 20 3.1 Iterationtrees. .... ... .... ... .... ... .. 22 3.2 Thecomparisonprocess . 27 4 TheDodd-JensenLemma. 32 4.1 Thecopyingconstruction . 32 4.2 TheDodd-JensenLemma . 37 4.3 The Weak Dodd-Jensen Property . 39 5 SolidityandCondensation . 42 6 Background-Certified Fine Extender Sequences . 49 6.1 Kc constructions ... ... .... ... .... ... .. 50 6.2 The iterability of Kc .................... 52 6.3 Large cardinals in Kc .................... 61 7 The reals of Mω ........................... 63 7.1 Iteration strategies in L(R) ................ 64 7.2 Correctness and genericity iterations . 69 R 8 HODL( ) belowΘ.......................... 80 3 4 CONTENTS I. An Outline of Inner Model Theory John R. Steel 1. Introduction This article is an exposition of the theory of canonical inner models for large cardinal hypotheses, or extender models. We hope to convey the most important ideas and methods of this theory without sinking into the morass of fine-structural detail surrounding them. The resulting outline should be accessible to anyone familiar with the theory of iterated ultrapowers and L[µ] contained in Kunen’s paper [14], and with the fine structure theory for L contained in Jensen’s paper [11]. We shall present basic inner model theory in what is roughly the greatest generality in which it is currently known. This means that the theory we shall outline applies to extender models which may satisfy large cardinal hypotheses as strong as “There is a Woodin cardinal which is a limit of Woodin cardinals”. -
Iteration Trees
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 7. Number I. January 1994 ITERATION TREES D. A. MARTIN AND J. R. STEEL O. INTRODUCTION In this paper we shall develop some ideas which lead to a proof of the fol- lowing Theorem. Suppose there are n Woodin cardinals, where n < w. Then there is a proper class model Mn such that Mn F ZFC + «There are n Woodin cardinals" 1 + "lR has a Lln+2 wellorder." This theorem is best possible, in the sense that if there are n Woodin cardi- . 1 . nals and a measurable cardinal above them all, then lR has no Lln+2 well order [ShW], and in fact I1~+1 determinacy holds [MS]. The theorem shows that the large cardinal hypothesis used to prove I1~+1 determinacy in [MS] cannot be substantially improved. Our theorem is an application of the theory of core models or canonical inner models for large cardinal hypotheses. Our work develops further some of the basic ideas of this theory. The theory itself has many other applications, particularly in the realm of relative consistency results. (As a quotable paradigm for a broad class of results, let us mention the following theorem of Solovay and Shelah: ZF + DC + "All sets of reals are Lebesgue measurable" is consistent if and only if ZFC + "There is an inaccessible cardinal" is consistent. In the "only if' direction, the model that one produces of ZFC + "There is an inaccessible cardinal" is a core model and, in fact, the smallest core model, Godel's L.) We shall begin with a general description of core model theory, of some of its history, and of the advance we have made. -
[Math.LO] 13 Jun 2012 Descriptive Inner Model Theory
Descriptive inner model theory ∗† Grigor Sargsyan ‡ Department of Mathematics Rutgers University Hill Center for the Mathematical Sciences 110 Frelinghuysen Rd. Piscataway, NJ 08854 USA http://math.rutgers.edu/∼gs481 [email protected] Dedicated to my father Edvard Sargsyan on the occasion of his 60th birthday June 14, 2012 Abstract The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One arXiv:1206.2712v1 [math.LO] 13 Jun 2012 particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. The program of constructing canonical inner models for large cardinals has been a source of increasingly sophisticated ideas leading to a beautiful theory of canonical models of fragments ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †Keywords: Mouse, inner model theory, descriptive set theory, hod mouse. ‡This material is partially based upon work supported by the National Science Foundation under Grant No DMS-0902628. -
Inner Model Theoretic Geology∗
Inner model theoretic geology∗ Gunter Fuchsy Ralf Schindler September 9, 2015 Abstract One of the basic concepts of set theoretic geology is the mantle of a model of set theory V: it is the intersection of all grounds of V, that is, of all inner models M of V such that V is a set-forcing extension of M. The main theme of the present paper is to identify situations in which the mantle turns out to be a fine structural extender model. The first main result is that this is the case when the universe is constructible from a set and there is an inner model with a Woodin cardinal. The second situation like that arises if L[E] is an extender model that is iterable in V but not internally iterable, as guided by P -constructions, L[E] has no strong cardinal, and the extender sequence E is ordinal definable in L[E] and its forcing extensions by collapsing a cutpoint to ! (in an appropriate sense). The third main result concerns the Solid Core of a model of set theory. This is the collection of all sets that are constructible from a set of ordinals that cannot be added by set-forcing to an inner model. The main result here is that if there is an inner model with a Woodin cardinal, then the solid core is a fine-structural extender model. 1 Introduction In [FHR15], the authors introduced several types of inner models which are defined following the paradigm of \undoing" forcing. Thus, the mantle M of a model of set theory V is the intersection of all of its ground models (i.e., the intersection of all inner models of which V is a set-generic forcing extension).