91 research outputs found

    Large Cardinals

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    Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (V=L)

    The real numbers in inner models of set theory

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    Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Joan BagariaWe study the structural regularities and irregularities of the reals in inner models of set theory. Starting with LL, Gödel's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of LL and its levels, mainly referring to [18]. We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to L[#1]L\left[\#_1\right] (the smallest inner model closed under the sharp operation for reals) and L[#]L[\#] (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models

    Large Cardinals, Inner Models, and Determinacy:An Introductory Overview

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    Preserving levels of projective determinacy by tree forcings

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    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

    On arbitrary sets and ZFC

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    Set theory deals with the most fundamental existence questions in mathematics– questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximalist. After explaining what is meant by definability and by “arbitrariness”, a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds too fferan elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.Junta de Andalucía P07-HUM-02594Ministerio de Ciencia y Tecnología BFF2003-09579-C0

    Thin collections of sets of projective ordinals and analogs of L

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    ON ABSOLUTELY MEASURABLE SETS

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    Generalised Hopficity and Products of the Integers

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    Hopfian groups have been a topic of interest in alge-braic settings for many years. In this work a natural generalizationof the notion, so-called R-Hopficity is introduced. Basic propertiesof R-Hopfian groups are developed and the question of whether ornot infinite direct products of copies of the integers are R-Hopfian isconsidered. An unexpected result is that the answer to this purelyalgebraic question depends on the set theory assumed
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