DOCSLIB.ORG

Corey Bacal Switzer

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Corey Bacal Switzer COREY BACAL SWITZER CONTACT 365 5th Avenue, New York, NY 1001 Department of Mathematics, The Graduate Center, CUNY [email protected] http:nncoreyswitzer.wordpress.com Phone: +1-917-742-4728 EDUCATION Ph.D. in Mathematics, May 2020 (Expected) The Graduate Center, The City University of New York Dissertation Advisors: Gunter Fuchs and Joel David Hamkins Dissertation: Alternative Cicho´nDiagrams and Forcing Axioms Compatible with CH Masters, Second Year in Mathematical Logic and Foundations of Computer Science, May 2016, Honors (Mention Bien) L'Universit´ede Paris 7, Denis Diderot, Paris, France Thesis Advisor: Boban Veliˇckovi´c Thesis: Iterating Generalized Proper Forcings with Side Conditions Masters, First Year in Logic, May 2015, High Honors (Mention Tr`esBien) L'Universit´ede Paris 1, Panth´eon-Sorbonne, Paris, France B.A. in Mathematics and Philosophy, May 2014, High Honors (First Class) Trinity College, Dublin, Ireland A.A. in Liberal Arts, May 2014, Obtained jointly with my high school diploma Bard College, New York PUBLICATIONS AND OTHER WRITING Journal Articles 1. Corey Switzer, The Cichon Diagram for Degrees of Constructibility, forthcoming in Math- ematical Logic Quarterly (2020). 2. Corey Switzer, Specializing Wide Aronszajn Trees Without Adding Reals, Mathematics ArXiv, arXiv:2002.08206 [math.LO], (2020), submitted to the RIMS Kokyuroku. 3. Corey Switzer, Higher Dimensional Cardinal Characteristics for Sets of Functions, Mathe- matics ArXiv, arXiv:1909.07458 [math.LO], (2019), Submitted to The Annals of Pure and Applied Logic. 4. Corey Switzer, Independence in Arithmetic: The Method of (L; n)-Models, Mathematics ArXiv. arXiv: 1906.04273 [math.LO], (2019), submitted to The Bulletin of Symbolic Logic. 5. Gunter Fuchs and Corey Switzer, Iterating Subversion Forcing Classes, in preparation, 30 Pages. 6. Joel David Hamkins and Corey Switzer, The Class Forcing Axiom, in preparation, 9 Pages. Theses 1. Corey Switzer, Alternative Cicho´nDiagrams and Forcing Axioms Compatible with CH, PhD Dissertation, CUNY Graduate Center, (2020). 2. Corey Switzer, Iterating Generalized Proper Forcings with Side Conditions, 2nd Year Mas- ters' thesis, L'Universit´ede Paris VII, (2016). 3. Corey Switzer, Grands Cardinaux, Mod`elesInt´erieurset un Probl`emede Woodin, 1st Year Masters' thesis, L'Universit´ede Paris I, (2015). RESEARCH INTERESTS Mathematical Logic, Set Theory, and Combinatorics: cardinal characteristics, iterated forcing, infinite combinatorics, forcing axioms, model theory of arithmetic, higher computability theory, Ramsey theory, graph theory, set theoretic topology and measure theory TEACHING Hunter College, CUNY Winter 2018{Present Graduate Teaching Fellow Winter 2018{Present As a graduate teaching fellow, I teach at Hunter College, CUNY. I have been the primary instructor for every course I have taught, in charge of writing and delivering lectures, producing syllabi, meeting with students, writing and grading exams and organizing all other aspects of the courses. My teaching has been well received by students. I currently have a 5.0 rating on ratemyprofessor.com, and similarly high scores on internal Hunter reviews. Courses Taught: Math 150 Calculus with Analytic Geometry I (Spring 2018 (2 sections), Summer 2018, Fall 2018 (2 Sections), Spring 2019, Fall 2019, Spring 2020) Math 156 Mathematical Proofs (Winter 2018) Math 125 Precalculus (Spring 2019) CONFERENCE TALKS Infinity Subversion Forcing, University of Kyoto, Japan (November 2019), RIMS Set The- ory Workshop 2019 Cardinal Characteristics for Sets of Functions, University of Kobe, Japan (November 2019) (L; n)-Models, CUNY Graduate Center (May 2019), Journ´eessur les Arithm´etiquesFaibles Models of SCFA Violating CH, University of Campinas, Brazil (December 2018), Cantor Meets Robinson The Cichon Diagram for Degrees of Constructibility, University of Wisconsin, Madison (March 2018), Graduate Student Logic Conference SEMINAR TALKS Generalized Cardinal Characteristics for Sets of Functions, Kurt G¨odelResearch Center, University of Vienna (January 2020), KGRC Research Seminar Eventual Domination on Baire Space, and Higher Dimensional Analogues, Virginia Com- monwealth University (October 2019), Analysis, Logic, Physics Seminar Constructions of Size Continuum, CUNY Graduate Center (October 2019), Models of Peano Arithmetic Seminar (L; n)-models, University of Pennsylvania (April 2019), Logic and Computation Seminar Independence in Arithmetic and Non-Standard Combinatorics, BCC CUNY (March 2019), BCC Math Seminar The Cichon Diagram for Degrees of Constructibility, Rutgers University (March 2019), Rutgers Logic Seminar Sets at Arm's Length, CUNY Graduate Center (February 2019), Models of Peano Arith- metic Seminar Why PA?, CUNY Graduate Center (Novermber 2018), Models of Peano Arithmetic Sem- inar Cichon Diagrams for Degree Theories, University of Wisconsin, Madison (November 2018), UW Logic Seminar Models of SCFA Violating CH, CUNY Graduate Center (September 2018), CUNY Set Theory Seminar Applications of (L; n)-Models, CUNY Graduate Center (September 2018), Models of Peano Arithmetic Seminar Forcing Over Arithmetic: Arboreal Forcings, CUNY Graduate Center (February 2018), Models of Peano Arithmetic Seminar A Cichon Diagram for Degrees of Constructibility CUNY Graduate Center (November 2017), CUNY Set Theory Seminar, this talk, which was on original research, served as my Oral Exam, for which I was given High Pass Iterating Proper and Semiproper Forcings with Side Conditions, CUNY Graduate Center, (April 2017), CUNY Set Theory Seminar The HOD Dichotomy, Rutgers University (March 2017), Learning Seminar on the HOD Conjecture Recursively Saturated, Rather Classless Models in the Constructible Universe, CUNY Grad- uate Center (March 2017), Models of Peano Arithmetic Seminar AWARDS AND FELLOWSHIPS CUNY Mathematics Fellowship, 5 Year Fully Funded PhD contract, CUNY Graduate Center, Fall 2016-Present ($ 255,000 over 5 years, 5th year projected to be declined due to finishing early) Invited Research Visit to the Kurt G¨odelResearch Center, Vienna, Austria, January 2020 Invited Research Visit to the University of Kobe, Japan, November 2019 Doctoral Students Research Grant, Awarded for the project Forcing Axioms Compatible with CH, CUNY Graduate Center, June 2019- June 2020 Received Funding to attend 50 Years of Set Theory in Toronto at The Fields Institue of Mathematical Sciences in Toronto, Canada, May 2019 Invited research visit to University College, Oxford, UK January 2019 Received funding from the Association of Symbolic Logic to attend the conference Cantor Meets Robinson, University of Campinas, December 2018 Received funding to attend Appalachian Set Theory at Carnegie Mellon University, June 2018 Received funding to attend Summer School in Descriptive Set Theory and Ergodic Theory, UC Irvine, August 2017 Recipient of the John Isaac Beare Prize for the best undergraduate philosophy papers at Trinity College, Dublin, Fall 2011 Recipient of the Louis Kennedy Prize for the highest exam scores in mathematics at Trinity College, Dublin, Fall 2011 Recipient of the Townsend Memorial Prize for Excellence in mathematics at Trinity College, Dublin, Fall 2011 SERVICE Co-organizer (with V. Gitman), Set Theory Seminar, The Graduate Center, CUNY. Fall 2018{Present. Co-organizer (with G. Fuchs and V. Gitman), CUNY Spring Fling Set Theory Conference, an NSF funded conference to be held at CUNY April 22-24, 2020. Organizer, Student Set Theory Seminar, The Graduate Center, CUNY. Fall 2018. LANGUAGES English Mother Tongue French Near Native Fluency Spanish Conversational REFERENCES Joel David Hamkins Professor in Logic, Faculty of Philosophy University of Oxford Email: [email protected] Gunter Fuchs Professor in Mathematics The Graduate Center and the College of Staten Island, CUNY Email: [email protected] Roman Kossak Professor in Mathematics The Graduate Center and BCC, CUNY Email: [email protected] (For Teaching) Ilya Kapovich Professor of Mathematics Hunter College, CUNY Email: [email protected].
Load more

Recommended publications
  • The Universal Finite Set 3
    THE UNIVERSAL FINITE SET JOEL DAVID HAMKINS AND W. HUGH WOODIN Abstract. We define a certain finite set in set theory { x | ϕ(x) } and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition ϕ has complexity Σ2, so that any affirmative instance of it ϕ(x) is verified in any sufficiently large rank-initial segment of the universe Vθ ; the set is empty in any transitive model and others; and if ϕ defines the set y in some countable model M of ZFC and y ⊆ z for some finite set z in M, then there is a top-extension of M to a model N in which ϕ defines the new set z. Thus, the set shows that no model of set theory can realize a maximal Σ2 theory with its natural number parameters, although this is possible without parameters. Using the universal finite set, we prove that the validities of top-extensional set-theoretic potentialism, the modal principles valid in the Kripke model of all countable models of set theory, each accessing its top-extensions, are precisely the assertions of S4. Furthermore, if ZFC is consistent, then there are models of ZFC realizing the top-extensional maximality principle. 1. Introduction The second author [Woo11] established the universal algorithm phenomenon, showing that there is a Turing machine program with a certain universal top- extension property in models of arithmetic.
    [Show full text]
  • Open Class Determinacy Is Preserved by Forcing 3
    OPEN CLASS DETERMINACY IS PRESERVED BY FORCING JOEL DAVID HAMKINS AND W. HUGH WOODIN Abstract. The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary trans- finite recursion ETRΓ for a fixed class well-order Γ is preserved by pre-tame class forcing. The full principle ETR itself is preserved by countably strate- gically closed pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Meanwhile, it remains open whether ETR is preserved by all forcing, including the forcing merely to add a Cohen real. 1. Introduction The principle of elementary transfinite recursion ETR—according to which every first-order class recursion along any well-founded class relation has a solution—has emerged as a central organizing concept in the hierarchy of second-order set theories from G¨odel-Bernays set theory GBC up to Kelley-Morse set theory KM and beyond. Many of the principles in the hierarchy can be seen as asserting that certain class recursions have solutions. KM + class-DC KM+CC KM 1 GBC + Π1-comprehension GBC + Open determinacy for class games GBC+ETR = GBC+ Clopen determinacy for class games = GBC + iterated truth predicates GBC + ETROrd = GBC+ Class forcing theorem arXiv:1806.11180v1 [math.LO] 28 Jun 2018 = GBC + truth predicates for LOrd,ω(∈, A) = GBC + truth predicates for LOrd,Ord(∈, A) = GBC + Ord-iterated truth predicates = GBC + Boolean set-completions exist = GBC+Clopen determinacy for class games of rank Ord+1 GBC + ETRω GBC + Con(GBC) GBC In addition, many of these principles, including ETR and its variants, are equiv- alently characterized as determinacy principles for certain kinds of class games.
    [Show full text]
  • The Multiverse Perspective on Determinateness in Set Theory
    The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse Axioms Multiverse Mathematics The multiverse perspective on determinateness in set theory Joel David Hamkins New York University, Philosophy The City University of New York, Mathematics College of Staten Island of CUNY The CUNY Graduate Center Exploring the Frontiers of Incompleteness Harvard University, October 19, 2011 The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse Axioms Multiverse Mathematics Introduction Set theory as Ontological Foundation Set theorists commonly take their subject to serve as a foundation for the rest of mathematics, in the sense that other abstract mathematical objects can be construed fundamentally as sets, and in this way, they regard the set-theoretical universe as the realm of all mathematics. On this view, mathematical objects—such as functions, groups, real numbers, ordinals—are sets, and being precise in mathematics amounts to specifying an object in set theory. A slightly weakened position remains compatible with structuralism, where the set-theorist holds that sets provide objects fulfilling the desired structural roles of mathematical objects, which therefore can be viewed as sets. The multiverse perspective, Harvard EFI 2011 Joel David Hamkins, New York The multiverse view Ontology of Forcing Case Studies: CH, V = L Multiverse Axioms Multiverse Mathematics Introduction The Set-Theoretical Universe These sets accumulate transfinitely to form the universe of all sets. This cumulative universe is regarded by set theorists as the domain of all mathematics. The orthodox view among set theorists thereby exhibits a two-fold realist or Platonist nature: First, mathematical objects (can) exist as sets, and Second, these sets enjoy a real mathematical existence, accumulating to form the universe of all sets.
    [Show full text]
  • Satisfaction Is Not Absolute 3
    SATISFACTION IS NOT ABSOLUTE JOEL DAVID HAMKINS AND RUIZHI YANG Abstract. We prove that the satisfaction relation N |= ϕ[~a] of first-order logic is not absolute between models of set theory having the structure N and the formulas ϕ all in common. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic hN, +, ·, 0, 1, <i, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and the same arithmetic truths, yet disagree on their truths-about-truth, at any desired level of the iterated truth-predicate hierarchy; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have the same hHω2 , ∈i or the same rank-initial segment hVδ, ∈i, yet disagree on which assertions are true in these structures. On the basis of these mathematical results, we argue that a philosophical commitment to the determinateness of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic struc- ture N = { 0, 1, 2,... } itself, but rather, we argue, is an additional higher-order commitment requiring its own analysis and justifica- tion. arXiv:1312.0670v1 [math.LO] 3 Dec 2013 The first author’s research has been supported in part by Simons Foundation grant 209252, by PSC-CUNY grant 66563-00 44 and by grant 80209-06 20 from the CUNY Collaborative Incentive Research Award program, and he is grateful for the support provided to him as a visitor to Fudan University in Shanghai in June, 2013, where much of this research took place.
    [Show full text]
  • Set Theory and Its Place in the Foundations of Mathematics: a New Look at an Old Question Mirna Džamonja
    Set Theory and its Place in the Foundations of Mathematics: A New Look at an Old Question Mirna Džamonja To cite this version: Mirna Džamonja. Set Theory and its Place in the Foundations of Mathematics: A New Look at an Old Question. Journal of Indian Council of Philosophical Research, 2017, 34 (2), pp.415-424. 10.1007/s40961-016-0082-6. hal-01858066 HAL Id: hal-01858066 https://hal.archives-ouvertes.fr/hal-01858066 Submitted on 19 Aug 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Set Theory and its Place in the Foundations of Mathematics- a new look at an old question Mirna Dˇzamonja ∗ (to appear in the Journal of Indian Council of Philosophical Research) Abstract This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one. 1 1 Introduction Set theory is often cited as the foundations of mathematics. In this paper we review the history of that claim, argue that the other known foundations such as category theory and univalent foundations have the same but not larger right to exists, and conclude that the only choice which agrees with the mathematical practice of today is to take a pluralist view of the matter and accept all these as foundations of an interesting part of mathematics, each in its own right.
    [Show full text]
  • Here High School Juniors and Seniors Take College Courses for Credit
    KAMERYN J. WILLIAMS [email protected] · kamerynjw.net Department of Mathematics · University of Hawai`i at M¯anoa 2565 McCarthy Mall, Keller 401A · Honolulu, HI 96822 ACADEMIC HISTORY University of Hawai`i at M¯anoa Fall 2018{Present Temporary Assistant Professor EDUCATION Ph.D. in Mathematics, May 2018 The Graduate Center, The City University of New York Dissertation Advisor: Joel David Hamkins (jdh.hamkins.org) Dissertation Title: The structure of models of second-order set theories B.S. in Mathematics, Dec 2012, High Honors Boise State University Minor in Computer Science RESEARCH INTERESTS I am a mathematical logician, specializing in set theory. My work has touched on the foundations and applications of class forcing; the model theory of sets; set-theoretic geology; and connections to philosophy of mathematics. PUBLICATIONS Publications in refereed journals. Jonas Reitz and Kameryn J. Williams, Inner mantles and iterated HOD, Mathematical Logic Quarterly, Vol 65, No. 4, pp. 498{510. (2019). DOI: doi:10.1002/malq.201800071. Preprint: arXiv:1810.08702 [math.LO]. Kameryn J. Williams, Minimum models of second-order set theories, Journal of Symbolic Logic, Vol 8, No. 2, pp. 589{620. (2019). DOI: doi:10.1017/jsl.2019.27. Preprint: arXiv:1709.03955 [math.LO]. Miha E. Habiˇc,Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner, and Kameryn J. Williams, Set-theoretic blockchains, Archive for Mathematical Logic, Vol. 58, No. 7{8, pp. 965{997. (2019). DOI: doi:10.1007/s00153-019-00672-z. Preprint: arXiv:1808.01509 [math.LO]. Publications to appear. Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlict, and Kameryn J.
    [Show full text]
  • Joel David Hamkins Education Academic Appointments
    Joel David Hamkins http://jdh.hamkins.org Education Ph.D. in Mathematics, May 1994, University of California, Berkeley C.Phil. in Mathematics, December 1991, University of California, Berkeley B.S. in Mathematics, May 1988, California Institute of Technology M.A. by Resolution, September 2018, Oxford University Academic Appointments University of Oxford, 2018{present Professor of Logic, Faculty of Philosophy, University of Oxford Affiliated Faculty, Mathematical Institute, University of Oxford Sir Peter Strawson Fellow in Philosophy, University College, Oxford The City University of New York, 1995{2018, with various leaves College of Staten Island of CUNY, Mathematics Full Professor since 2003, tenure granted 2000 Associate Professor 1999-2002, Assistant Professor 1995-1998 The Graduate Center of CUNY Doctoral Faculty in Philosophy since 2013 Doctoral Faculty in Computer Science since 2002 Doctoral Faculty in Mathematics since 1997 Isaac Newton Institute of Mathematical Sciences, Cambridge, U.K. Visiting Fellow, August{October, 2015 Visiting Fellow, March{April, June, 2012 New York University Visiting Professor of Philosophy, January{June, 2015 Visiting Professor of Philosophy, July{December, 2011 Fields Institute, University of Toronto Scientific Researcher, August, 2012 University of Vienna, Kurt G¨odelResearch Center, Guest Professor, June, 2009 Universiteit van Amsterdam, Institute for Logic, Language & Computation Visiting Professor, April{August 2007 NWO Bezoekersbeurs visiting researcher, June{August 2005, June 2006 Universit¨atM¨unster,Institut
    [Show full text]
  • Arxiv:1210.6541V1 [Math.LO] 24 Oct 2012 Uy2,2018 27, July E Hoit Fe Ru Gis H Xo Fconstructibility of Axiom the Against Argue Often Theorists Set Introduction 1
    July 27, 2018 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in Multiverse˙on˙V=L A multiverse perspective on the axiom of constructiblity Joel David Hamkinsa Visiting Professor of Philosophy, New York University Professor of Mathematics, The City University of New York The Graduate Center & College of Staten Island [email protected], http://jdh.hamkins.org I shall argue that the commonly held V =6 L via maximize position, which rejects the axiom of constructibility V = L on the basis that it is restrictive, implicitly takes a stand in the pluralist debate in the philosophy of set theory by presuming an absolute background concept of ordinal. The argument appears to lose its force, in contrast, on an upwardly extensible concept of set, in light of the various facts showing that models of set theory generally have extensions to models of V = L inside larger set-theoretic universes. 1. Introduction Set theorists often argue against the axiom of constructibility V = L on the arXiv:1210.6541v1 [math.LO] 24 Oct 2012 basis that it is restrictive. Some argue that we have no reason to think that every set should be constructible, or as Shelah puts it, “Why the hell should it be true?” [19]. To suppose that every set is constructible is seen as an artificial limitation on set-theoretic possibility, and perhaps it is a mistaken principle generally to suppose that all structure is definable. Furthermore, aThis article expands on an argument that I made during my talk at the Asian Initiative for Infinity: Workshop on Infinity and Truth, held July 25–29, 2011 at the Institute for Mathematical Sciences, National University of Singapore.
    [Show full text]
  • Same Structure, Different Truth
    Philosophy math Satisfaction is not absolute Consequences and further examples A few gems Conclusion Same structure, different truth Joel David Hamkins City University of New York CUNY Graduate Center Mathematics, Philosophy, Computer Science College of Staten Island Mathematics MathOverflow CSLI 2016 Workshop on Logic, Rationality & Intelligent Interaction Stanford University Same structure, different truth, CSLI 2016 Joel David Hamkins, New York Philosophy math Satisfaction is not absolute Consequences and further examples A few gems Conclusion This is joint work with Ruizhi Yang of Fudan University, Shanghai. Our paper is available on my blog: J. D. Hamkins, R. Yang, “Satisfaction is not absolute,” to appear in the Review of Symbolic Logic. Same structure, different truth, CSLI 2016 Joel David Hamkins, New York Philosophy math Satisfaction is not absolute Consequences and further examples A few gems Conclusion My favorite situation A philosophical concern... leads to interesting mathematical questions... whose answers illuminate the philosophical issue. Same structure, different truth, CSLI 2016 Joel David Hamkins, New York Philosophy math Satisfaction is not absolute Consequences and further examples A few gems Conclusion My favorite situation A philosophical concern... leads to interesting mathematical questions... whose answers illuminate the philosophical issue. Same structure, different truth, CSLI 2016 Joel David Hamkins, New York Philosophy math Satisfaction is not absolute Consequences and further examples A few gems Conclusion
    [Show full text]
  • The Set-Theoretic Multiverse
    THE REVIEW OF SYMBOLIC LOGIC Volume 5, Number 3, September 2012 THE SET-THEORETIC MULTIVERSE JOEL DAVID HAMKINS Department of Philosophy, New York University, Mathematics Program, The Graduate Center of The City University of New York, and Department of Mathematics, The College of Staten Island of CUNY Abstract. The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set- theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. §1. Introduction. Set theorists commonly take their subject as constituting an ontological foundation for the rest of mathematics, in the sense that abstract mathematical objects can be construed fundamentally as sets, and because of this, they regard set theory as the domain of all mathematics. Mathematical objects are sets for them, and being precise in mathematics amounts to specifying an object in set theory. These sets accumulate transfinitely to form the cumulative universe of all sets, and the task of set theory is to discover its fundamental truths.
    [Show full text]
  • Inner Models with Large Cardinal Features Usually Obtained by Forcing
    INNER MODELS WITH LARGE CARDINAL FEATURES USUALLY OBTAINED BY FORCING ARTHUR W. APTER, VICTORIA GITMAN, AND JOEL DAVID HAMKINS Abstract. We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver inde- structible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly com- pact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the mea- surable cardinals are bounded below it and another inner model W with a κ HV W strongly compact cardinal , such that κ+ ⊆ HOD . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level in- equivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals.
    [Show full text]
  • Kameryn J Williams
    KAMERYN J WILLIAMS [email protected] · kamerynblog.wordpress.com Ph.D. Program in Mathematics · The Graduate Center, The City University of New York 365 Fifth Avenue, New York, NY 10016 EDUCATION Ph.D. in Mathematics, (Expected) May 2018 The Graduate Center, The City University of New York Dissertation Advisor: Joel David Hamkins (jdh.hamkins.org) Dissertation Title: The structure of models of second-order set theories B.S. in Mathematics, Dec 2012, High Honors Boise State University Minor in Computer Science Overall GPA: 3.852 PUBLICATIONS Kameryn J. Williams, The Structure of Models of Second-Order Set Theories, Ph.D. Dis- sertation, The City University of New York, eprint: arXiv:1804.09526 [math.LO] (2018). Joel David Hamkins, Philip Welch, Kameryn Williams, The universal finite sequnece of ordinals in the constructible universe, in preparation. Miha Habiˇc, Joel David Hamkins, Lukas Daniel Klausner, Jonathan Verner, and Kameryn J. Williams, Set-theoretic blockchains, in preparation. Kameryn J. Williams, Least models of second-order set theories, under review, eprint: arXiv:1709.03955 [math.LO] (2017). Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlict, and Kameryn Williams, The exact strength of the class forcing theorem, under review, eprint: arXiv:1707.03700 [math.LO] (2017). CONFERENCE TALKS On the length of iterated full satisfaction classes, Warsaw Workshop on Formal Truth Theories, Institute of Philosophy, University of Warsaw (Sept 2017). Minimal models of Kelley-Morse set theory, Boise Extravaganza in Set Theory, part of the 97th Annual Meeting of the AAAS Pacific Division, University of San Diego (June 2016). Minimal models of Kelley-Morse set theory, NY Graduate Student Logic Conference, The Graduate Center, CUNY.
    [Show full text]