12 research outputs found

    A polychromatic Ramsey theory for ordinals

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    The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω ω . We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In this version the ordinal and the colouring are presentable by finite automata and the clique is additionally required to be regular. The corresponding automatic Ramsey degree turns out to be greater than the set theoretic Ramsey degree. Finally, we demonstrate that a version for computable structures fails

    Separating club-guessing principles in the presence of fat forcing axioms

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    We separate various weak forms of Club Guessing at ω1\omega_1 in the presence of 202^{\aleph_0} large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with ω\omega-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds 1\aleph_1-many reals but preserves CH

    Set Theory

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    This meeting covered all important aspects of modern Set Theory, including large cardinal theory, combinatorial set theory, descriptive set theory, connections with algebra and analysis, forcing axioms and inner model theory. The presence of an unusually large number (19) of young researchers made the meeting especially dynamic

    The model-theoretic complexity of automatic linear orders

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    Automatic structures are—possibly infinite—structures which are finitely presentable by means of finite automata on strings or trees. Largely motivated by the fact that their first-order theories are uniformly decidable, automatic structures gained a lot of attention in the "logic in computer science" community during the last fifteen years. This thesis studies the model-theoretic complexity of automatic linear orders in terms of two complexity measures: the finite-condensation rank and the Ramsey degree. The former is an ordinal which indicates how far a linear order is away from being dense. The corresponding main results establish optimal upper bounds on this rank with respect to several notions of automaticity. The Ramsey degree measures the model-theoretic complexity of well-orders by means of the partition relations studied in combinatorial set theory. This concept is investigated in a purely set-theoretic setting as well as in the context of automatic structures.Auch im Buchhandel erhältlich: The model-theoretic complexity of automatic linear orders / Martin Huschenbett Ilmenau : Univ.-Verl. Ilmenau, 2016. - xiii, 228 Seiten ISBN 978-3-86360-127-0 Preis (Druckausgabe): 16,50

    Forcing consequences of PFA together with the continuum large

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    We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a large continuum.Comment: 35 page

    Forcing with Adequate Sets of Models as Side Conditions

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    We present a general framework for forcing on ω2\omega_2 with finite conditions using countable models as side conditions. This framework is based on a method of comparing countable models as being membership related up to a large initial segment. We give several examples of this type of forcing, including adding a function on ω2\omega_2, adding a nonreflecting stationary subset of ω2cof(ω)\omega_2 \cap \textrm{cof}(\omega), and adding an ω1\omega_1-Kurepa tree

    Coherent Adequate Sets and Forcing Square

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    We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω2\omega_2 using finite conditions
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