Symmetric Models, Singular Cardinal Patterns, and Indiscernibles

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 ℵω. Here we show how to construct several patterns 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 ℵρ+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 equiconsistency of the existence of the ω1-Erdös cardinal with the original Chang conjecture, we prove equiconsistencies for some variants of Chang conjectures in models of ZF with various forms of Erdös cardinals in models of ZFC. Such equiconsistency results 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

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Combinatorial properties and dependent choice in symmetric extensions based on Lévy collapse

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of ZFC, then DC<κ can be preserved in the symmetric extension of V in terms of symmetric system ⟨ P, G, F⟩ , if P is κ-distributive and F is κ-complete. Further we observe that if δ< κ and V is a model of ZF+ DCδ, then DCδ can be preserved in the symmetric extension of V in terms of symmetric system ⟨ P, G, F⟩ , if P is (δ+ 1)-strategically closed and F is κ-complete. © 2022, The Author(s)

Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms

This dissertation explores the justification of strong theories of sets extending Zeremelo-Fraenkel set theory with choice and large cardinal axioms. In particular, there are two noted program providing axioms extending this theory: the inner model program and the forcing axiom program. While these programs historically developed to serve different mathematical goals and ends, proponents of each have attempted to justify their preferred axiom candidate on the basis of its supposed maximization potential. Since the maxim of ‘maximize’ proves central to the justification of ZFC+LCs itself, and shows up centrally in the current debate over how to best extend this theory, any attempt to resolve this debate will need to investigate the relationship between maximization notions and the candidates for a strong theory of sets. This dissertation takes up just this project.The first chapter of this dissertation describes the history of axiom selection in set theory, focusing on developments since 1980 which have led to the two standard axiom candidates for extending ZFC+LCs: V = Ult(L) and Martin’s Maximum. The second chapter explains the justification of the methodological maxim of ‘maximize’ as an informal principle, and presents two formal explications of the notion: one due to John Steel, the other to Penelope Maddy. The third chapter directly examines whether either approach to axioms can be truly said to maximize over the other. It is shown that the axiom candidates are equivalent in Steel’s sense of ‘maximize’, while in Maddy’s sense of ‘maximize’, Martin’s Maximum is found to maximize over V = Ult(L). Given the strong justification of Maddy’s explication in terms of the goals of set theory as a foundational discipline, it is argued that this result raises a serious justificatory challenge for advocates of the inner model program. The fourth chapter considers future directions of research, focusing on possible responses to the justificatory challenge, and highlighting issues that must be overcome before a full justificatory story of forcing axioms can be developed

The core model induction in a choiceless context

In der Arbeit wird Woodin's Methode der Kernmodellinduktion benutzt, um die relative Konsistenz des Determiniertheitsaxiom zu zeigen. Dabei wird von einem Modell von ZF ausgegangen in dem das Auswahlaxiom nicht erfüllt ist und gezeigt, dass es ein Modell von ZF gibt in dem das Determiniertheitsaxiom gilt. Genauer werden folgende Resultate gezeigt: (1) Angenommen V ist ein Modell von "ZF + alle überabzählbaren Nachfolgerkardinalzahlen sind schwach kompakt und alle überabzählbaren Limeskardinalzahlen sind singulär". Dann gilt AD^L(R) in einer generischen Erweiterung von HOD_X. (2) Angenommen V ist ein Modell von "ZF + alle überabzählbaren Kardinalzahlen sind singulär". Dann gilt AD^L(R) in einer generischen Erweiterung von HOD_X

Preserving levels of projective determinacy by tree forcings

We prove that various classical tree forcings -- for instance Sacks forcing, Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve the statement that every real has a sharp and hence analytic determinacy. We then lift this result via methods of inner model theory to obtain level-by-level preservation of projective determinacy (PD). Assuming PD, we further prove that projective generic absoluteness holds and no new equivalence classes classes are added to thin projective transitive relations by these forcings.Comment: 3 figure

Stationary set preserving L-forcings and the extender algebra

Wir konstruieren das Jensensche L-Forcing und nutzen dieses um die Pi_2 Konsequenzen der Theorie ZFC+BMM+"das nichtstationäre Ideal auf omega_1 ist abschüssig" zu studieren. Viele natürliche Konsequenzen der Theorie ZFC+MM folgen schon aus dieser schwächeren Theorie. Wir geben eine neue Charakterisierung des Axioms Dagger ("Alle Forcings welche stationäre Teilmengen von omega_1 bewahren sind semiproper") in dem wir eine Klasse von L-Forcings isolieren deren Semiproperness äquivalent zu Dagger ist. Wir verallgemeinern ein Resultat von Todorcevic: wir zeigen, dass Rado's Conjecture Dagger impliziert. Des weiteren studieren wir Generizitätsiterationen im Kontext einer messbaren Woodinzahl. Mit diesem Werkzeug erhalten wir eine Verallgemeinerung des Woodinschen Sigma^2_1 Absolutheitstheorems. We review the construction of Jensen's L-forcing which we apply to study the Pi_2 consequences of the theory ZFC + BMM + "the nonstationary ideal on omega_1 is precipitous". Many natural consequences ZFC + MM follow from this weaker theory. We give a new characterization of the axiom dagger ("All stationary set preserving forcings are semiproper") by isolating a class of stationary set preserving L-forcings whose semiproperness is equivalent to dagger. This characterization is used to generalize work of Todorcevic: we show that Rado's Conjecture implies dagger. Furthermore we study genericity iterations beginning with a measurable Woodin cardinal. We obtain a generalization of Woodin's Sigma^2_1 absoluteness theorem

Views from a peak:Generalisations and descriptive set theory

This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality

A Potpourri of Partition Properties

The cardinal characteristic inequality r <= hm3 is proved. Several partition relations for ordinals and one for countable scattered types are given. Moreover partition relations for lexicographically ordered sequences of zeros and ones are given in a no-choice context