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  1. Uncountable structures are not classifiable up to bi-embeddability.Filippo Calderoni, Heike Mildenberger & Luca Motto Ros - 2019 - Journal of Mathematical Logic 20 (1):2050001.
    Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic164(12) (2013) 1454–1492], we show that whenever κ is a cardinal satisfying κ<κ=κ>ω, then the embeddability relation between κ-sized structures is strongly invariantly universal, and hence complete for (κ-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or (...)
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  • A generalized Borel-reducibility counterpart of Shelah’s main gap theorem.Tapani Hyttinen, Vadim Kulikov & Miguel Moreno - 2017 - Archive for Mathematical Logic 56 (3-4):175-185.
    We study the κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document}, if T is classifiable and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is not, then the isomorphism of models of T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
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  • Universality for Orders and Graphs Which Omit Large Substructures.Katherine Thompson - 2006 - Notre Dame Journal of Formal Logic 47 (2):233-248.
    This paper will examine universality spectra for relational theories which cannot be described in first-order logic. We will give a method using functors to show that two types of structures have the same universality spectrum. A combination of methods will be used to show universality results for certain ordered structures and graphs. In some cases, a universal spectrum under GCH will be obtained. Since the theories are not first-order, the classic model theory result under GCH does not hold.
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  • Kurepa trees and spectra of $${mathcal {L}}{omega 1,omega }$$ L ω 1, ω -sentences.Dima Sinapova & Ioannis Souldatos - 2020 - Archive for Mathematical Logic 59 (7-8):939-956.
    We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \-sentence \ that codes Kurepa trees to prove the following statements: The spectrum of \ is consistently equal to \ and also consistently equal to \\), where \ is weakly inaccessible.The amalgamation spectrum of \ is consistently equal to \ and \\), where again \ is weakly inaccessible. This is the first example of an \-sentence whose spectrum and amalgamation spectrum are consistently both right-open and (...)
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  • Chain models, trees of singular cardinality and dynamic ef-games.Mirna Džamonja & Jouko Väänänen - 2011 - Journal of Mathematical Logic 11 (1):61-85.
    Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf. With a notion of satisfaction and -isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game (...)
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  • Club Guessing and the Universal Models.Mirna Džamonja - 2005 - Notre Dame Journal of Formal Logic 46 (3):283-300.
    We survey the use of club guessing and other PCF constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal, element.
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  • Closed Maximality Principles and Generalized Baire Spaces.Philipp Lücke - 2019 - Notre Dame Journal of Formal Logic 60 (2):253-282.
    Given an uncountable regular cardinal κ, we study the structural properties of the class of all sets of functions from κ to κ that are definable over the structure 〈H,∈〉 by a Σ1-formula with parameters. It is well known that many important statements about these classes are not decided by the axioms of ZFC together with large cardinal axioms. In this paper, we present other canonical extensions of ZFC that provide a strong structure theory for these classes. These axioms are (...)
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  • Σ1(κ)-definable subsets of H.Philipp Lücke, Ralf Schindler & Philipp Schlicht - 2017 - Journal of Symbolic Logic 82 (3):1106-1131.
    We study Σ1-definable sets in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1-definable, the set of all stationary subsets of ω1 is not Σ1-definable and the complement of every Σ1-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1-definable well-ordering (...)
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  • Questions on generalised Baire spaces.Yurii Khomskii, Giorgio Laguzzi, Benedikt Löwe & Ilya Sharankou - 2016 - Mathematical Logic Quarterly 62 (4-5):439-456.
    We provide a list of open problems in the research area of generalised Baire spaces, compiled with the help of the participants of two workshops held in Amsterdam (2014) and Hamburg (2015).
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  • On ‐complete equivalence relations on the generalized Baire space.Tapani Hyttinen & Vadim Kulikov - 2015 - Mathematical Logic Quarterly 61 (1-2):66-81.
    Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if, then many of them are ‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is ‐complete (it is, if, but can be forced not to be).
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  • Regularity properties on the generalized reals.Sy David Friedman, Yurii Khomskii & Vadim Kulikov - 2016 - Annals of Pure and Applied Logic 167 (4):408-430.
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  • Large cardinals and definable well-orders, without the GCH.Sy-David Friedman & Philipp Lücke - 2015 - Annals of Pure and Applied Logic 166 (3):306-324.
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