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Symmetric Models, Singular Cardinal Patterns, and Indiscernibles Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Rheinischen Friedrich-Wilhelms-Universit¨atBonn vorgelegt von Ioanna Matilde Dimitriou aus Thessaloniki Bonn, 2011 2 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at Rheinischen Friedrich-Wilhelms-Universit¨atBonn 1. Gutachter: Prof. Dr. Peter Koepke 2. Gutachter: Prof. Dr. Benedikt L¨owe Tag der Promotion: 16. Dezember 2011 Erscheinungsjahr: 2012 3 Summary This thesis is on the topic of set theory and in particular large cardinal axioms, singular cardinal patterns, and model theoretic principles in models of set theory without the axiom of choice (ZF). The first task is to establish a standardised setup for the technique of symmetric forcing, our main tool. This is handled in Chapter 1. Except just translating the method in terms of the forcing method we use, we expand the technique with new definitions for properties of its building blocks, that help us easily create symmetric models with a very nice property, i.e., models that satisfy the approximation lemma. Sets of ordinals in symmetric models with this property are included in some model of set theory with the axiom of choice (ZFC), a fact that enables us to partly use previous knowledge about models of ZFC in our proofs. After the methods are established, some examples are provided, of constructions whose ideas will be used later in the thesis. The first main question of this thesis comes at Chapter 2 and it concerns patterns of singular cardinals in ZF, also in connection with large cardinal axioms. When we do assume the axiom of choice, every successor cardinal is regular and only certain limit cardinals are singular, such as @!. Here we show how to construct almost any pattern of singular and regular cardinals in ZF. Since the partial orders that are used for the constructions of Chapter 1 cannot be used to construct successive singular cardinals, we start by presenting some partial orders that will help us achieve such combinations. The techniques used here are inspired from Moti Gitik's 1980 paper \All uncountable cardinals can be singular", a straightforward modification of which is in the last section of this chapter. That last section also tackles the question posed by Arthur Apter \Which cardinals can become simultaneously the first measurable and first regular uncountable cardinal?". Most of this last part is submitted for publication in a joint paper with Arthur Apter , Peter Koepke, and myself, entitled \The first measurable and first regular cardinal can simultaneously be @ρ+1, for any ρ". Throughout the chapter we show that several large cardinal axioms hold in the symmetric models we produce. The second main question of this thesis is in Chapter 3 and it concerns the consistency strength of model theoretic principles for cardinals in models of ZF, in connection with large cardinal axioms in models of ZFC. The model theoretic principles we study are variations of Chang conjectures, which, when looked at in models of set theory with choice, have very large consistency strength or are even inconsistent. We found that by removing the axiom of choice their consistency strength is weakened, so they become easier to study. Inspired by the proof of the equiconsis- tency of the existence of the !1-Erd}oscardinal with the original Chang conjecture, we prove equiconsistencies for some variants of Chang conjectures in models of ZF with various forms of Erd}oscardinals in models of ZFC. Such equiconsistency re- sults are achieved on the one direction with symmetric forcing techniques found in Chapter 1, and on the converse direction with careful applications of theorems from core model theory. For this reason, this chapter also contains a section where the most useful `black boxes' concerning the Dodd-Jensen core model are collected. More detailed summaries of the contents of this thesis can be found in the beginnings of Chapters 1, 2, and 3, and in the conclusions, Chapter 4. 4 Curriculum Vitae Ioanna Matilde Dimitriou Education 1/2006 - 12/2011 Mathematisches Institut der Universit¨atBonn PhD student, subject: set theory supervisor: Prof.Dr.Peter Koepke 9/2003 - 12/2005 Institute for Logic, Language and Computation (ILLC), Universiteit van Amsterdam Master of Science, subject: logic Thesis: \Strong limits and inaccessibility with non wellorderable owersets", supervisor: Prof.Dr.Benedikt L¨owe 10/1998 - 7/2003 Department of Mathematics, Aristotle University of Thessaloniki, Greece. Bachelor degree in Mathematics. Scientific Publications Ioanna Dimitriou, Peter Koepke, and Arthur Apter. The first measurable cardinal can be the first uncountable regular cardinal at any successor height. submitted for publication Andreas Blass, Ioanna M. Dimitriou, Benedikt L¨owe. Inaccessible cardinals with- out the axiom of choice. Fundamenta Mathematicae, vol.194, pp. 179-189 L. Afanasiev, P. Blackburn, I. Dimitriou, B. Gaiffe, E. Goris, M. Marx and M. de Rijke. PDL for ordered trees. Journal of Applied Non-Classical Logic (2005), 15(2): 115-135 Teaching experience Tutor and substitute for the lecture \Higher set theory: Formal derivations and natural proofs". Winter semester 2010-11, Univesit¨atBonn. Leader for the ILLC Master's research project \Singularizing successive cardinals". June 2009, Institute for Logic, Language, and Computation, Univestiteit van Am- sterdam Graduate seminar on logic (with P.Koepke and J.Veldman). Winter semester 2008- 09, Univestit¨atBonn 5 Graduate Seminar on Logic (with P.Koepke). Summer semester 2008, Universit¨at Bonn Tutor for the class \Set theory I" (Mengenlehre I). Winter semester 2006-07, Uni- versit¨atBonn. Carer for the \Diplomandenseminar" (with P.Koepke). Summer semester 2006, Universit¨atBonn. Tutor for the class \Einf¨uhrungin die Mathematische Logik". Summer semes- ter 2006, Universit¨atBonn. Carer for Blockseminar \Mengenlehre; Large cardinals and determinacy" (with P.Koepke and B.L¨owe) February 11-12, 2006, Amsterdam. Invited talks Amsterdam workshop in set theory. \The first uncountable regular can be the first measurable". Amsterdam, 26-27 May 2009 Oberseminar of the mathematical logic group of M¨unster. \A modern approach to Gitiks All uncountable cardinals can be singular". M¨unster,19 June 2008. Logic seminar of the logic group of Leeds. “L¨owenheim-Skolem type properties in set theory". Leeds, 7 May 2008. 1 Games in Logic, Language and Computation 14 2 meeting. \Topological regulari- ties in second order arithmetic". Amsterdam, 28 September 2007. Oberseminar of the mathematical logic group of Bonn. \Fragments of Choice under the axiom of Determinacy". June 2, 2005, Bonn, Germany. Other professional activities Organising committee for the \Young set theory workshop 2011", plus website and booklet design. K¨onigswinter (near Bonn), 21-25 March 2011 Website and booklet design for the \Young set theory workshop 2010". Raach (near Vienna), February 15 - 19, 2010 Organisor of the first workshop of the project \ICWAC: Infinitary combinatorics without the axiom of choice", plus website and booklet design. Bonn, 11-13 June 2009. Organising committee of the \Young set theory workshop 2008", and editor of 6 the workshop report. Bonn, 21-25 January 2008. Scientific secretary of \BIWOC: Bonn International Workshop on Ordinal Com- putability", plus website and booklet design. Organising committee of the \Colloquium Logicum 2006" conference. September 22-24, 2006, Bonn, Germany. Employment November 15, 2006 to February 28, 2011 scientific staff (wissenschaftliche Mitar- beiterin) at the Mathematics department of University Bonn. January 2, 2006 to November 14, 2006, scientific assistant (wissenschaftliche Hilf- skraft) at the Mathematics department of University of Bonn. Languages Greek: native English: fluent German: advanced Spanish: intermediate Dutch: beginner 7 Acknowledgements Doctoral theses take a long time to prepare, and this process is not as smooth as one might hope or plan. With this note I would like to thank the people who played a crucial role in this process. I would like to thank first and foremost my parents Christos and Marta, for their loving support, also financial but mainly emotional, and for the safety net that they provide me with. An essential role in my emotional stability played also my partner Calle Runge, whom I thank from the bottom of my heart for his emotional support, for providing me with a homely environment to escape to, and for our amazing conversations about life, the universe, and everything. Also when stress got way too much, he was there to help me, along with Jonah Lamp and Mark Jackson, who with me are \Jonah Gold and his Silver Apples". Guys, thank you so much for all the fun and stress relief that drumming in a band gave me in the difficult times of a PhD. I am grateful to my supervisor Peter Koepke for the opportunity to fulfill this dream of mine, and for all that he taught me, not only in the realm of core model theory, but also in the social aspects of the academic world. A big thank you goes also to Benedikt L¨owe, the second referee of this thesis, for his invaluable advice ever since my master's studies at the ILLC in Amsterdam, and for his question on patterns of singular cardinals that led to Section 2 of Chapter 2. Special thanks go to Andr´esCaicedo and to my officemate Thilo Weinert, for their suggestions and corrections of earlier versions of the thesis, and a big thanks to the Mathematical logic group of Bonn for listening to my detailed presentations of Chapters 1 and 2. I also thank Arthur Apter for coauthoring a paper with Peter Koepke and myself, and for the informative discussions. I am grateful to all the people in the \Young set theory workshop" for the engaging talks, formal and informal. I would like to express my appreciation to Moti Gitik whose ideas, especially in his paper \All uncountable cardinals can be singular" have kept me busy for a good two years during my PhD studies, and have taught me so much.
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