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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 . -
Faculty Document 2436 Madison 7 October 2013
University of Wisconsin Faculty Document 2436 Madison 7 October 2013 MEMORIAL RESOLUTION OF THE FACULTY OF THE UNIVERSITY OF WISCONSIN-MADISON ON THE DEATH OF PROFESSOR EMERITA MARY ELLEN RUDIN Mary Ellen Rudin, Hilldale professor emerita of mathematics, died peacefully at home in Madison on March 18, 2013. Mary Ellen was born in Hillsboro, Texas, on December 7, 1924. She spent most of her pre-college years in Leakey, another small Texas town. In 1941, she went off to college at the University of Texas in Austin, and she met the noted topologist R.L. Moore on her first day on campus, since he was assisting in advising incoming students. He recognized her talent immediately and steered her into the math program, which she successfully completed in 1944, and she then went directly into graduate school at Austin, receiving her PhD under Moore’s supervision in 1949. After teaching at Duke University and the University of Rochester, she joined the faculty of the University of Wisconsin-Madison as a lecturer in 1959 when her husband Walter came here. Walter Rudin died on May 20, 2010. Mary Ellen became a full professor in 1971 and professor emerita in 1991. She also held two named chairs: she was appointed Grace Chisholm Young Professor in 1981 and Hilldale Professor in 1988. She received numerous honors throughout her career. She was a fellow of the American Academy of Arts and Sciences and was a member of the Hungarian Academy of Sciences, and she received honorary doctor of science degrees from the University of North Carolina, the University of the South, Kenyon College, and Cedar Crest College. -
Axiomatic Set Teory P.D.Welch
Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. -
Real Proofs of Complex Theorems (And Vice Versa)
REAL PROOFS OF COMPLEX THEOREMS (AND VICE VERSA) LAWRENCE ZALCMAN Introduction. It has become fashionable recently to argue that real and complex variables should be taught together as a unified curriculum in analysis. Now this is hardly a novel idea, as a quick perusal of Whittaker and Watson's Course of Modern Analysis or either Littlewood's or Titchmarsh's Theory of Functions (not to mention any number of cours d'analyse of the nineteenth or twentieth century) will indicate. And, while some persuasive arguments can be advanced in favor of this approach, it is by no means obvious that the advantages outweigh the disadvantages or, for that matter, that a unified treatment offers any substantial benefit to the student. What is obvious is that the two subjects do interact, and interact substantially, often in a surprising fashion. These points of tangency present an instructor the opportunity to pose (and answer) natural and important questions on basic material by applying real analysis to complex function theory, and vice versa. This article is devoted to several such applications. My own experience in teaching suggests that the subject matter discussed below is particularly well-suited for presentation in a year-long first graduate course in complex analysis. While most of this material is (perhaps by definition) well known to the experts, it is not, unfortunately, a part of the common culture of professional mathematicians. In fact, several of the examples arose in response to questions from friends and colleagues. The mathematics involved is too pretty to be the private preserve of specialists. -
Recognizable Sets and Woodin Cardinals: Computation Beyond the Constructible Universe
RECOGNIZABLE SETS AND WOODIN CARDINALS: COMPUTATION BEYOND THE CONSTRUCTIBLE UNIVERSE MERLIN CARL, PHILIPP SCHLICHT AND PHILIP WELCH Abstract. We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. • Computable sets are elements of L, while recognizable objects appear up to the level of Woodin cardinals. • A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M 1, 1 where M denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ. Contents 1. Introduction1 2. Definitions and basic facts3 3. Recognizable sets5 3.1. Machines with fixed ordinal parameters6 3.2. Different ordinals9 3.3. Arbitrary ordinals 10 3.4. The recognizable closure and HOD 13 4. Recognizable sets and M1 14 4.1. Subsets of countable ordinals 14 4.2. Subsets of !1 17 4.3. Structures which are not recognizable 18 4.4. Generically recognizable sets 20 5. Conclusion and questions 20 References 21 1. -
Small Cardinals and Small Efimov Spaces 3
SMALL CARDINALS AND SMALL EFIMOV SPACES WILL BRIAN AND ALAN DOW Abstract. We introduce and analyze a new cardinal characteristic of the continuum, the splitting number of the reals, denoted s(R). This number is connected to Efimov’s problem, which asks whether every infinite compact Hausdorff space must contain either a non-trivial con- vergent sequence, or else a copy of βN. 1. Introduction This paper is about a new cardinal characteristic of the continuum, the splitting number of the reals, denoted s(R). Definition 1.1. If U and A are infinite sets, we say that U splits A provided that both A ∩ U and A \ U are infinite. The cardinal number s(R) is defined as the smallest cardinality of a collection U of open subsets of R such that every infinite A ⊆ R is split by some U ∈U. In this definition, R is assumed to have its usual topology. The number s(R) is a topological variant of the splitting number s, and is a cardinal characteristic of the continuum in the sense of [2]. Most of this paper is devoted to understanding the place of s(R) among the classical cardinal characteristics of the continuum. Our main achievement along these lines is to determine completely the place of s(R) in Cichoń’s diagram. More precisely, for every cardinal κ appearing in Cichoń’s diagram, we prove either that κ is a (consistently strict) lower bound for s(R), or that κ is a (consistently strict) upper bound for s(R), or else that each of κ< s(R) and s(R) < κ is consistent. -
Ronald Jensen Receives a 2003 Steele Prize Gottfried Wilhelm-Leibniz-Preisträger 2003
Allyn Jackson Ronald Jensen receives a 2003 Steele Prize At the 109th Annual Meeting of the AMS in Baltimore in January 2003, a Steele Prize has been awarded to Ronald Jensen for a Seminal Contribution to Research. The text that follows presents the selection committee’s citation and a brief biographical sketch. (Allyn Jackson) Ronald Jensen’s paper,“The fine structure of the con- of an inner model, in this case G¨odel’s constructible structible hierarchy” (Annals of Mathematical Logic sets, that go far beyond G¨odel’s proof of the general- 4 (1972) 229–308), has been of seminal importance ized continuum hypothesis in this model. for two different directions of research in contempo- The second direction initiated by Jensen’s paper rary set theory: the inner model program and the use involves applying these combinatorial principles to of combinatorial principles of the sort that Jensen es- problems arising in other parts of mathematics. The tablished for the constructible universe. principle, which Jensen proved to hold in the con- The inner model program, one of the most active structible universe, has been particularly useful in parts of set theory nowadays, has as its goals the un- such applications. By now, it has become part of the derstanding of very large cardinals and their use to standard tool kit of several branches of mathematics, measure the consistency strength of assertions about ranging from general topology to module theory. much smaller sets. A central ingredient of this pro- Ronald Jensen received his Ph.D. in 1964 from the gram is to build, for a given large cardinal axiom, a University of Bonn. -
Squares and Uncountably Singularized Cardinals
SQUARES AND UNCOUNTABLY SINGULARIZED CARDINALS MAXWELL LEVINE AND DIMA SINAPOVA Abstract. It is known that if κ is inaccessible in V and W is an outer model of + V + W W V such that (κ ) = (κ ) and cf (κ) = !, then κ,ω holds in W . (Many strengthenings of this theorem have been investigated as well.) We show that this theorem does not generalize to uncountable cofinalities: There is a model V in which κ is inaccessible, such that there is a forcing extension W of V in + V + W W which (κ ) = (κ ) and ! < cf (κ) < κ, while in W , κ,τ fails for all τ < κ. We make use of Magidor's forcing for singularizing an inaccessible κ to have uncountable cofinality. Along the way, we analyze stationary reflection in this model, and we show that it is possible for κ,cf(κ) to hold in a forcing extension by Magidor's poset if the ground model is prepared with a partial square sequence. 1. Introduction Singular cardinals are a topic of great interest in set theory, largely because they are subject to both independence results and intricate ZFC constraints. Here we focus on the subject of singularized cardinals: cardinals that are regular in some inner model and singular in some outer model. Large cardinals are in fact necessary to consider such situations, so in actuality we consider cardinals that are inaccessible in some inner model and singular in some outer model. Singularized cardinals were originally sought in order to prove the consistency of the failure of the Singular Cardinals Hypothesis|the idea was to blow up the powerset of a cardinal and then singularize it|and obtaining singularized cardinals was a significant problem in its own right. -
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: -
Arxiv:1710.03586V1 [Math.LO] 10 Oct 2017 Strong Compactness and the Ultrapower Axiom
Strong Compactness and the Ultrapower Axiom Gabriel Goldberg October 11, 2017 1 Some consequences of UA We announce some recent results relevant to the inner model problem. These results involve the development of an abstract comparison theory from a hy- pothesis called the Ultrapower Axiom, a natural generalization to larger car- dinals of the assumption that the Mitchell order on normal measures is linear. Roughly, it says that any pair of ultrapowers can be ultrapowered to a com- mon ultrapower. The formal statement is not very different. Let Uf denote the class of countably complete ultrafilters. MU Ultrapower Axiom. For any U0, U1 ∈ Uf, there exist W0 ∈ Uf 0 and MU W1 ∈ Uf 1 such that M M U0 U1 MW0 = MW1 and M M U0 U1 jW0 ◦ jU0 = jW1 ◦ jU1 The Ultrapower Axiom, which we will refer to from now on as UA, holds in all known inner models. This is a basic consequence of the methodology used to build these models, the comparison process. For example, assuming there arXiv:1710.03586v1 [math.LO] 10 Oct 2017 is a proper class of strong cardinals and V = HOD, UA follows from Woodin’s principle Weak Comparison [1], which is an immediate consequence of the basic comparison lemma used to construct and analyze canonical inner models. These hypotheses are just an artifact of the statement of Weak Comparison, and in fact the methodology of inner model theory simply cannot produce models in which UA fails (although of course it can produce models in which V =6 HOD and there are no strong cardinals). -
The Semi-Weak Square Principle
The Semi-Weak Square Principle Maxwell Levine Universit¨atWien Kurt G¨odelResearch Center for Mathematical Logic W¨ahringerStraße 25 1090 Wien Austria [email protected] Abstract Cummings, Foreman, and Magidor proved that for any λ < κ, κ,λ implies that ∗ κ carries a PCF-theoretic object called a very good scale, but that κ (which one could write as κ,κ) is consistent with the absence of a very good scale at κ. They asked whether κ,<κ is enough to imply the existence of a very good scale for κ, and we resolve this question in the negative. Furthermore, we sharpen a theorem of Cummings and Schimmerling and show that κ,<κ implies the failure of simultaneous stationary reflection at κ+ for any singular κ. This implies as a corollary that if Martin's Maximum holds, then κ,<κ fails for singular cardinals κ of cofinality !1. Keywords: Set Theory, Forcing, Large Cardinals 1. Background Singular cardinals occupy a curious place in set theory. They are amenable to the forcing technique devised by Cohen to establish the independence of the continuum hypothesis, but many of their properties can be proved outright from ZFC using the PCF theory developed by Shelah. The properties of singular cardinals in a given model depend on a fundamental tension between the extent to which the model supports large cardinals, and the extent to which it resembles G¨odel'sconstructible universe L. The square principle, denoted κ if it holds at a cardinal κ, was distilled by Jensen in order to find Suslin trees at successors of singular cardinals in L [8], and it embodies many of the constructions that can be carried out in L. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title The Combinatorics and Absoluteness of Definable Sets of Real Numbers Permalink https://escholarship.org/uc/item/0v60v6m4 Author Norwood, Zach Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles ¿e Combinatorics and Absoluteness of Denable Sets of Real Numbers A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Zach Norwood 2018 © Copyright by Zach Norwood 2018 ABSTRACT OF THE DISSERTATION ¿e Combinatorics and Absoluteness of Denable Sets of Real Numbers by Zach Norwood Doctor of Philosophy in Mathematics University of California, Los Angeles, 2018 Professor Itay Neeman, Chair ¿is thesis divides naturally into two parts, each concerned with the extent to which the theory of LR can be changed by forcing. ¿e rst part focuses primarily on applying generic-absoluteness principles to show that denable sets of reals enjoy regularity properties. ¿e work in Part I is joint with Itay Neeman and is adapted from our forthcoming paper [33]. ¿is project was motivated by questions about mad families, maximal families of innite subsets of ω any two of which have only nitely many members in common. We begin, in the spirit of Mathias [30], by establishing (¿eorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem [48] that there are no innite mad families in the Solovay model. In Chapter3 we stray from the main line of inquiry to briey study a game-theoretic characteri- zation of lters with the Baire Property.