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7ContentsIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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6From the editorsIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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5Preface – Unity and Diversity of LogicIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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70Reseña de "Filosofía sintética de las matemáticas contemporáneas" de Fernando ZalameaIdeas Y Valores 59 (142): 174-182. 2010.
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64The small index property for homogeneous models in AEC’sArchive for Mathematical Logic 57 (1-2): 141-157. 2018.We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah. We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.
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139Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics (edited book)De Gruyter. 2015.In recent years, mathematical logic has developed in many directions, the initial unity of its subject matter giving way to a myriad of seemingly unrelated areas. The articles collected here, which range from historical scholarship to recent research in geometric model theory, squarely address this development. These articles also connect to the diverse work of Väänänen, whose ecumenical approach to logic reflects the unity of the discipline.
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91Uniqueness of limit models in classes with amalgamationMathematical Logic Quarterly 62 (4-5): 367-382. 2016.We prove the following main theorem: Let be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim‐Skolem number of the class. If is μ‐Galois‐stable, has no μ‐Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two ‐limits over M, for, are isomorphic over M.
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127Around Logical PerfectionTheoria 87 (4): 971-985. 2021.In this article we present a notion of “logical perfection”. We first describe through examples a notion oflogical perfectionextracted from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notio…Read more
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52Preface – Unity and Diversity of LogicIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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202Chains of end elementary extensions of models of set theoryJournal of Symbolic Logic 63 (3): 1116-1136. 1998.Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (`unfoldable cardinals') lie in the boundary of the propositions consistent with `V = L' and the existence of 0 ♯ . We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various forcing constructions
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48A Radio Interview with Jouko VäänänenIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. pp. 417-422. 2015.
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63ContentsIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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51The Hart-Shelah example, in stronger logicsAnnals of Pure and Applied Logic 172 (6): 102958. 2021.
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120Toward categoricity for classes with no maximal modelsAnnals of Pure and Applied Logic 97 (1-3): 1-25. 1999.We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these class…Read more
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16From the editorsIn Åsa Hirvonen, Juha Kontinen, Roman Kossak & Andrés Villaveces (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. 2015.
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81Around independence and domination in metric abstract elementary classes: assuming uniqueness of limit modelsMathematical Logic Quarterly 60 (3): 211-227. 2014.We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability‐like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs.
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143Limit models in metric abstract elementary classes: the categorical caseMathematical Logic Quarterly 62 (4-5): 319-334. 2016.We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability-like assumptions, we generalize some theorems from 7, 15-17. We prove criteria for existence and uniqueness of limit models in the metric context.
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88Categoricity, External and Internal: An Excerpt from a Conversation with Saharon ShelahTheoria 87 (4): 1001-1012. 2021.A long series of conversations interweaving mathematical, historical and philosophical aspects of categoricity in model theory took place between the author and Saharon Shelah in 2016 and 2017. In this excerpt of that long conversation, we explore the relationship between explicit and implicit aspects of categoricity. We also discuss the connection with definability issues.
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348Heights of Models of ZFC and the Existence of End Elementary Extensions IIJournal of Symbolic Logic 64 (3): 1111-1124. 1999.The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further investigate the connection between the height of M and the existence of End Elementary Extensions of M. In particular, we prove that the theory `ZFC + GCH + there exist measurable cardinals + all inaccessible non weakly compact cardinals are possible heights of models with no End Elementary Extensions' is …Read more
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440The notion of Mathematics as Ontology (as defined by Badiou in his work) is brought into question from a working mathematician's perspective. Notions of independence in set theory and model theory are contrasted with the original equation Mathematics=Ontology. The author builds an extension of mathematical ontology from set theory to a foliated, étale, setting.
