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  1. Fallen Cardinals.Menachem Kojman & Saharon Shelah - 2001 - Annals of Pure and Applied Logic 109 (1-2):117-129.
    We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra contains a complete subalgebra which is isomorphic to the collapse algebra CompCol. Consequently, adding a generic filter to the quotient algebra collapses μ0 to 1. Another corollary is that the Baire number of the space U of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.
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  • Toward Classifying Unstable Theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
  • On ◁∗-Maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.
    This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...)
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  • On the Existence of Universal Models.Mirna Džamonja & Saharon Shelah - 2004 - Archive for Mathematical Logic 43 (7):901-936.
    Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind (...)
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  • On Properties of Theories Which Preclude the Existence of Universal Models.Mirna Džamonja & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 139 (1):280-302.
    We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to (...)
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  • The Saturation of Club Guessing Ideals.Tetsuya Ishiu - 2006 - Annals of Pure and Applied Logic 142 (1):398-424.
    We prove that it is consistent that there exists a saturated tail club guessing ideal on ω1 which is not a restriction of the non-stationary ideal. Two proofs are presented. The first one uses a new forcing axiom whose consistency can be proved from a supercompact cardinal. The resulting model can satisfy either CH or 20=2. The second one is a direct proof from a Woodin cardinal, which gives a witnessing model with CH.
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  • The Descriptive Set-Theoretical Complexity of the Embeddability Relation on Models of Large Size.Luca Motto Ros - 2013 - Annals of Pure and Applied Logic 164 (12):1454-1492.
    We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...)
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