Symmetry in abstract elementary classes with amalgamation

2017

Arch. Math. Logic

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Abstract.Theorem 0.1. Let K be an abstract elementary class (AEC) with amalgamation and no maximal models. Let λ > LS(K). If K is categorical in λ, then the model of cardinality λ is Galois-saturated.This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: K has a unique limit model in each cardinal below λ, (when λ is big-enough) K is weakly tame below λ, and the thresholds of several existing categoricity transfers can be improved.We also prove a downward transfer of solvability (a version of superstability introduced by Shelah):Corollary 0.2. Let K be an AEC with amalgamation and no maximal models. Let λ > µ > LS(K). If K is solvable in λ, then K is solvable in µ.

Section: Remark 52 It Is Natural To Ask What Happens If µ = Ls(k) mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Saturation and solvability in abstract elementary classes with amalgamation

2017

Arch. Math. Logic

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“…Definition ; it can be shown that any reasonable forking‐like notion must be μ‐forking over saturated models [, 9.7]). In the setup of Corollary , it was known that μ‐forking satisfies all the conditions there except (3) (for symmetry, this is a recent result of the author [, 5.7(1)], relying on joint work with VanDieren ).…”

Section: Introductionmentioning

confidence: 98%

“…Fifth remark: if we add to the assumptions of Corollary that Galois types over saturated models of size μ are determined by their restrictions to model of size χ, for some

χ < μ

(this is called weak tameness in the literature), then the conclusion is known (cf. [, 6.4] and [, 5.7(1)]) and one can strengthen (5) to (5+), i.e., one gets a good μ‐frame. It is known how to derive eventual weak tameness from categoricity in a high‐enough cardinal, thus the conclusion also holds if μ is “high‐enough” (

μ \geq ℶ_{{false(2^{LS false(boldK false)} false)}^{+}}

suffices) [, 5.7(5)].…”

Section: Introductionmentioning

confidence: 99%

On the uniqueness property of forking in abstract elementary classes

Mathematical Logic Qtrly

2017

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In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.

Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

Mathematical Logic Qtrly

2018

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A new case of Shelah's eventual categoricity conjecture is established: Let scriptK be an abstract elementary class with amalgamation. Write μ:=ℶfalse(2 LS false(scriptKfalse)false)+ and H2:=ℶfalse(2μfalse)+. Assume that scriptK is H2‐tame and K≥H2 has primes over sets of the form M∪{a}. If scriptK is categorical in some λ>H2, then scriptK is categorical in all λ′≥H2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first‐order theory T and D is categorical in a λ>|T|, then D is categorical in all λ′≥minfalse(λ,ℶ(2|T|)+false).