05021 Abstracts Collection -- Mathematics, Algorithms, Proofs

From 09.01.05 to 14.01.05, the Dagstuhl Seminar 05021 ``Mathematics, Algorithms, Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. LinkstFo extended abstracts or full papers are provided, if available

Dependent choice, properness, and generic absoluteness

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas

Set Theory with Urelements

This dissertation aims to provide a comprehensive account of set theory with urelements. In Chapter 1, I present mathematical and philosophical motivations for studying urelement set theory and lay out the necessary technical preliminaries. Chapter 2 is devoted to the axiomatization of urelement set theory, where I introduce a hierarchy of axioms and discuss how ZFC with urelements should be axiomatized. The breakdown of this hierarchy of axioms in the absence of the Axiom of Choice is also explored. In Chapter 3, I investigate forcing with urelements and develop a new approach that addresses a drawback of the existing machinery. I demonstrate that forcing can preserve, destroy, and recover the axioms isolated in Chapter 2 and discuss how Boolean ultrapowers can be applied in urelement set theory. Chapter 4 delves into class theory with urelements. I first discuss the issue of axiomatizing urelement class theory and then explore the second-order reflection principle with urelements. In particular, assuming large cardinals, I construct a model of second-order reflection where the principle of limitation of size fails.Comment: arXiv admin note: text overlap with arXiv:2212.13627. Definition 15 in the previous versions is flawed, which is fixed in this versio

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

Is there a set of reals not in K(R)?

AbstractWe show, using the fine structure of K(R), that the theory ZF + AD + ∃X ⊆ R[X ∉ K(R)] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK(R) = K(P) for some P ⊆ (Θ+)K(R) where K(P) is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF + AD + ¬DCR implies that R† (dagger) exists

Competitive optimisation on timed automata

Timed automata are finite automata accompanied by a finite set of real-valued variables called clocks. Optimisation problems on timed automata are fundamental to the verification of properties of real-time systems modelled as timed automata, while the control-program synthesis problem of such systems can be modelled as a two-player game. This thesis presents a study of optimisation problems and two-player games on timed automata under a general heading of competitive optimisation on timed automata. This thesis views competitive optimisation on timed automata as a multi-stage decision process, where one or two players are confronted with the problem of choosing a sequence of timed moves—a time delay and an action—in order to optimise their objectives. A solution of such problems consists of the “optimal” value of the objective and an “optimal” strategy for each player. This thesis introduces a novel class of strategies, called boundary strategies, that suggest to a player a symbolic timed move of the form (b, c, a)— “wait until the value of the clock c is in very close proximity of the integer b, and then execute a transition labelled with the action a”. A distinctive feature of the competitive optimisation problems discussed in this thesis is the existence of optimal boundary strategies. Surprisingly perhaps, many competitive optimisation problems on timed automata of practical interest admit optimal boundary strategies. For example, optimisation problems with reachability price, discounted price, and average-price objectives, and two-player turn-based games with reachability time and average time objectives. The existence of optimal boundary strategies allows one to work with a novel abstraction of timed automata, called a boundary region graph, where players can use only boundary strategies. An interesting property of a boundary region graph is that, for every state, the set of reachable states is finite. Hence, the existence of optimal boundary strategies permits us to reduce competitive optimisation problem on a timed automaton to the corresponding competitive optimisation problem on a finite graph

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