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Upward Stability Transfer for Tame Abstract Elementary Classes

2006
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Notre Dame Journal of Formal Logic
*

Let K be an AEC

doi:10.1305/ndjfl/1153858652
fatcat:vttsmrxnfnexnlvgjy2xrkg2zy
*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. ...##
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Upward Stability Transfer for Tame Abstract Elementary Classes
[article]

2005
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arXiv
*
pre-print

We prove, for instance, that for tame

arXiv:math/0511748v1
fatcat:xhkrbqbdwnhhdejpchwz3gc5zi
*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. ...##
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The primal framework II: smoothness

1991
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Annals of Pure and Applied Logic
*

In a forthcoming paper, entitled, '

doi:10.1016/0168-0072(91)90095-4
fatcat:5grjfof2ercnng7jls5kxfif6m
*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 ...##
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The primal framework. II. Smoothness
[article]

1991
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arXiv
*
pre-print

This is the second in a series of articles developing

arXiv:math/9201246v1
fatcat:jlenojpebrg77lkvsvvloubtse
*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 ...##
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μ-Abstract elementary classes and other generalizations

2016
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Journal of Pure and Applied Algebra
*

We introduce μ-

doi:10.1016/j.jpaa.2016.02.002
fatcat:usqr6k5fpvccreg75v2dwgrefi
*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 . ...##
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Locally compact groups and continuous logic
[article]

2013
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arXiv
*
pre-print

We study expressive power of continuous logic in

arXiv:1206.5473v3
fatcat:4mfcljgxnnblpdmchochod3rsu
*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. ...##
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Logicality and Model Classes
[article]

2021
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arXiv
*
pre-print

We relate this to model-theoretic characteristics of

arXiv:2106.13506v2
fatcat:2kvfqj5dnjgj7f2cs6vermoznm
*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 ...##
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On categoricity in successive cardinals
[article]

2020
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arXiv
*
pre-print

We investigate, in ZFC, the behavior of

arXiv:1810.10061v4
fatcat:thwufgzu4recdipzdzed6is2ca
*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? ...##
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Structural logic and abstract elementary classes with intersections

2019
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Bulletin of the Polish Academy of Sciences Mathematics
*

As a corollary, we obtain that any AEC

doi:10.4064/ba8178-12-2018
fatcat:yummhk4alze5nfi3nwpwhtbuoe
*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). ...##
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Logicality and Model Classes

2021
*
Bulletin of Symbolic Logic
*

We relate this to modeltheoretic characteristics of

doi:10.1017/bsl.2021.42
fatcat:rkhmk3za7bfzfg7ilxjbzpjkeu
*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. ...##
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An Overview of Saharon Shelah's Contributions to Mathematical Logic, in Particular to Model Theory

2020
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Theoria
*

Back to Model Theory: A New Lindström Theorem Per Lindström (1969) proved that first-order logic is maximal

doi:10.1111/theo.12238
fatcat:xnglhrzxpjcdxfhfjuuw6smseu
*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 ...##
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Even ordinals and the Kunen inconsistency
[article]

2021
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arXiv
*
pre-print

By a recent result of Schlutzenberg, an

arXiv:2006.01084v3
fatcat:z6akrgsxfbdxxjusx3hpnhln5a
*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. ...##
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Algebraic description of limit models in classes of abelian groups
[article]

2019
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arXiv
*
pre-print

We study limit models in the

arXiv:1810.02203v6
fatcat:cbku2rdstzho5n5p3hfkojudpa
*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. ...##
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Actions of metric groups and continuous logic
[article]

2017
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arXiv
*
pre-print

We study expressive power of continuous logic in

arXiv:1706.04157v2
fatcat:7347gatyf5g45fqj7333eu6y6i
*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*cofinality*≤*ω*; ...##
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Internal sizes in μ-abstract elementary classes

2019
*
Journal of Pure and Applied Algebra
*

Working in the context of μ-

doi:10.1016/j.jpaa.2019.02.004
fatcat:zx2yfpxetvfgblfvzbd2mdb7ma
*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. ...
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