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Upward Stability Transfer for Tame Abstract Elementary Classes

John Baldwin, David Kueker, Monica VanDieren
2006 Notre Dame Journal of Formal Logic  
Let K be an AEC with Löwenheim-Skolem number ≤ κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then, K is Galois-stable in κ +n for all n < ω.  ...  Let K be an AEC satisfying the amalgamation property and with Löwenheim-Skolem number ℵ 0 that is ω-local and ℵ 0 -tame. If K is ℵ0-Galois-stable then K is Galois-stable in all cardinalities.  ...  Let K be an abstract elementary class that has Löwenheim-Skolem number ≤ κ and satisfies the amalgamation property and is κ-weakly tame.  ... 
doi:10.1305/ndjfl/1153858652 fatcat:vttsmrxnfnexnlvgjy2xrkg2zy

Upward Stability Transfer for Tame Abstract Elementary Classes [article]

John Baldwin, David Kueker, Monica VanDieren
2005 arXiv   pre-print
We prove, for instance, that for tame abstract elementary classes satisfying the amlagamation property and for large enough cardinals kappa, stability in kappa implies stability in kappa^+n for each natural  ...  number n.  ...  Let K be an abstract elementary class that has Löwenheim-Skolem number ≤ κ and satisfies the amalgamation property and is κ-weakly tame.  ... 
arXiv:math/0511748v1 fatcat:xhkrbqbdwnhhdejpchwz3gc5zi

The primal framework II: smoothness

J.T. Baldwin, S. Shelah
1991 Annals of Pure and Applied Logic  
In a forthcoming paper, entitled, 'Abstract classes with few models have 'homogeneous-universal' models', we show how to drop this predicate from the set of basic notions and still obtain results analogous  ...  This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains.  ...  In the first order case, the upwards Löwenheim-Skolem property is deduced from the compactness theorem; the downwards Löwenheim-Skolem property holds by the ability to form elementary submodels by adding  ... 
doi:10.1016/0168-0072(91)90095-4 fatcat:5grjfof2ercnng7jls5kxfif6m

The primal framework. II. Smoothness [article]

John T. Baldwin, Saharon Shelah
1991 arXiv   pre-print
This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains.  ...  It deals with the problem of smoothness and establishing the existence and uniqueness of a 'monster model'. We work here with a predicate for a canonically prime model.  ...  In the first order case, the upwards Löwenheim-Skolem property is deduced from the compactness theorem; the downwards Löwenheim-Skolem property holds by the ability to form elementary submodels by adding  ... 
arXiv:math/9201246v1 fatcat:jlenojpebrg77lkvsvvloubtse

μ-Abstract elementary classes and other generalizations

Will Boney, Rami Grossberg, Michael Lieberman, Jiří Rosický, Sebastien Vasey
2016 Journal of Pure and Applied Algebra  
We introduce μ-Abstract Elementary Classes (μ-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory.  ...  are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for μ-AECs transfer directly to accessible categories with  ...  Following 4.4 and 4.5, any µ-abstract class from 2.2 with (3) weakened to the existence of a weak Löwenheim-Skolem-Tarski number λ is a µ-AEC with LS(K) ≤ (λ + ) <µ .  ... 
doi:10.1016/j.jpaa.2016.02.002 fatcat:usqr6k5fpvccreg75v2dwgrefi

Locally compact groups and continuous logic [article]

Aleksander Ivanov
2013 arXiv   pre-print
We study expressive power of continuous logic in classes of (locally compact) groups. We also describe locally compact groups which are separably categorical structures.  ...  By the Löwenheim-Skolem theorem for countable fragments of L ω 1 ω ( [15] , p.69) any subset C of such a structure is contained in an elementary submodel of cardinality |C| (the countable fragment which  ...  Thus by the Löwenheim-Skolem theorem (in a 1-sorted language) non-compacness is bountiful in the class of locally compact groups.  ... 
arXiv:1206.5473v3 fatcat:4mfcljgxnnblpdmchochod3rsu

Logicality and Model Classes [article]

Juliette Kennedy, Jouko Väänänen
2021 arXiv   pre-print
We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi.  ...  We investigate which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier  ...  McGee points out that we could take a class size disjunction of L ∞∞ sentences and obtain a single 'sentence' which works in all cardinalities. As he points  ... 
arXiv:2106.13506v2 fatcat:2kvfqj5dnjgj7f2cs6vermoznm

On categoricity in successive cardinals [article]

Sebastien Vasey
2020 arXiv   pre-print
We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals.  ...  We prove for example that a universal L_ω_1, ω sentence categorical on an end segment of cardinals below _ω must be categorical also everywhere above _ω.  ...  Assume K is an AEC with Löwenheim-Skolem-Tarski number λ, categorical in every cardinal in [λ, ω (λ)). Must K be categorical everywhere above λ?  ... 
arXiv:1810.10061v4 fatcat:thwufgzu4recdipzdzed6is2ca

Structural logic and abstract elementary classes with intersections

Will Boney, Sebastien Vasey
2019 Bulletin of the Polish Academy of Sciences Mathematics  
As a corollary, we obtain that any AEC with countable Löwenheim-Skolem number is axiomatizable in L_∞, ω (Q), where Q is the quantifier "there exists uncountably many".  ...  We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for isomorphism types that we call structural logic: we prove that  ...  Thus we find that any AEC with intersections and countable Löwenheim-Skolem-Tarski number is axiomatizable in L ∞,ω (Q) (see Corollary 3.12).  ... 
doi:10.4064/ba8178-12-2018 fatcat:yummhk4alze5nfi3nwpwhtbuoe

Logicality and Model Classes

Juliette Kennedy, Jouko Väänänen
2021 Bulletin of Symbolic Logic  
We relate this to modeltheoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46] .  ...  We investigate which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier  ...  In particular: • ∆(L ∞ω ), even Σ(L ∞ω ), has a strong Löwenheim-Skolem theorem.  ... 
doi:10.1017/bsl.2021.42 fatcat:rkhmk3za7bfzfg7ilxjbzpjkeu

An Overview of Saharon Shelah's Contributions to Mathematical Logic, in Particular to Model Theory

Jouko Väänänen
2020 Theoria  
Back to Model Theory: A New Lindström Theorem Per Lindström (1969) proved that first-order logic is maximal with respect to a Downward Löwenheim-Skolem Theorem and the Compactness Theorem.  ...  For κ = ℶ κ the logic L 1 κ is maximal logic with respect to a Downward Löwenheim-Skolem property and the property of not being able (in a strong sense) to define the concept of well-ordering.  ...  The same happens with each f i . A model class K is said to be definable in L 1 κ if (roughly) there are a θ < κ and an α < κ such that if A ∈ K and player II has a winning strategy in the above game  ... 
doi:10.1111/theo.12238 fatcat:xnglhrzxpjcdxfhfjuuw6smseu

Even ordinals and the Kunen inconsistency [article]

Gabriel Goldberg
2021 arXiv   pre-print
By a recent result of Schlutzenberg, an elementary embedding from V_λ+2 to V_λ+2 does not suffice.  ...  The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding from V_λ+3 to V_λ+3 implies  ...  Suppose j : V λ+2 → V λ+2 is an elementary embedding with λ = κ ω (j). Suppose λ is a limit of Löwenheim-Skolem cardinals.  ... 
arXiv:2006.01084v3 fatcat:z6akrgsxfbdxxjusx3hpnhln5a

Algebraic description of limit models in classes of abelian groups [article]

Marcos Mazari-Armida
2019 arXiv   pre-print
We study limit models in the class of abelian groups with the subgroup relation and in the class of torsion-free abelian groups with the pure subgroup relation.  ...  the class of torsion-free abelian groups with the pure subgroup relation, then: * If the length of the chain has uncountable cofinality, then G (⊕_λQ ) ⊕Π_p prime(⊕_λZ_(p)). * If the length of the chain  ...  A good place to look for new classes of limit models is [BET07] . Definition 1 . 1 . 11 An abstract elementary class is a pair K = (K, ≤ K ), where: Date: August 20, 2019.  ... 
arXiv:1810.02203v6 fatcat:cbku2rdstzho5n5p3hfkojudpa

Actions of metric groups and continuous logic [article]

Aleksander Ivanov
2017 arXiv   pre-print
We study expressive power of continuous logic in classes of metric groups defined by properties of their actions. For example we consider properties non-OB, non-FH and non-FR.  ...  By the Löwenheim-Skolem theorem for countable fragments of L ω 1 ω ( [19] , p.69) any subset C of such a structure is contained in an elementary submodel of cardinality |C| (the countable fragment which  ...  Proposition 4.6 The following classes of groups are reducts of axiomatizable classes in L ω 1 ω : (1) The complement of the class of strongly bounded groups; (2) The class of groups of cofinalityω;  ... 
arXiv:1706.04157v2 fatcat:7347gatyf5g45fqj7333eu6y6i

Internal sizes in μ-abstract elementary classes

Michael Lieberman, Jiří Rosický, Sebastien Vasey
2019 Journal of Pure and Applied Algebra  
Working in the context of μ-abstract elementary classes (μ-AECs) - or, equivalently, accessible categories with all morphisms monomorphisms - we examine the two natural notions of size that occur, namely  ...  We also establish preliminary results on the existence and categoricity spectra of μ-AECs, including specific examples showing dramatic failures of the eventual categoricity conjecture (with categoricity  ...  We note that (by the L ω,ω Löwenheim-Skolem-Tarski theorem) M can be obtained as a λ-directed union of elementary substructures of cardinality strictly less than λ.  ... 
doi:10.1016/j.jpaa.2019.02.004 fatcat:zx2yfpxetvfgblfvzbd2mdb7ma
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