Forking in short and tame abstract elementary classes

2016

Arch. Math. Logic

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal $\kappa$. We show: $\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<\kappa)$-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: $\mathbf{Theorem}$ Let $K$ be a $\text{LS}(K)$-tame AEC with amalgamation. The following are equivalent: * $K$ is Galois stable in some $\lambda \ge \text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $\mu$ and $\lambda_0$ with $\mu \le \lambda_0 < \beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\lambda \ge \lambda_0$ with $\lambda = \lambda^{<\mu}$. $\mathbf{Theorem}$ Let $K$ be a fully $(<\kappa)$-tame and type short AEC with amalgamation, $\kappa = \beth_{\kappa} > \text{LS} (K)$. If $K$ is Galois stable, then the class of $\kappa$-Galois saturated models of $K$ admits an independence notion ($(<\kappa)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.Comment: 34 pages (v1 was split into this paper and arXiv:1503.01366

Section: Introductionmentioning

confidence: 99%

Infinitary stability theory

2016

Arch. Math. Logic

“…One such framework is Shelah's good λ-frames [She09, Section II.6]. Another is given by the definition of independence relation in [BGKV,Definition 3.1] (itself adapted from [BG,Definition 3.3] which can be traced back to the work of Makkai and Shelah [MS90]). Both definitions describe a relation "p does not fork over M" for p a Galois type over N and M ≤ N and require it to satisfy some properties.…”

Section: Independence Relationsmentioning

confidence: 99%

“…In particular, it has full model continuity and κ α (i) = α + + ℵ 0 . Full model continuity is not discussed in [MS90,BG], and κ α (i) = α + + ℵ 0 is only proven when α < κ or α = α <κ (see [BG,Theorem 8.2 .(3)]). Further, the proof uses the large cardinal axiom whereas we use it only to prove that coheir has the extension property.…”

Section: Applicationsmentioning

confidence: 99%

“…For completeness, we recall the definition of the following variation on the (< κ)-order property of length κ that appears in [BG,Definition 4.2] (but is adapted from a previous definition of Shelah, see there for more background):…”

Section: Preliminariesmentioning

confidence: 99%

“…Assume K is categorical in a λ > LS(K). Then: [BG,Theorem 6.6] Assume K does not have the weak κ-order property (see Definition 2.20) and LS(K) < κ ≤ µ < λ. Then: The next proposition is folklore: it derives joint embedding and no maximal models from amalgamation and categoricity.…”

Section: Shelah Claims In Chapter IV Of His Book That Solvabilitymentioning

confidence: 99%

See 2 more Smart Citations

Building independence relations in abstract elementary classes

Annals of Pure and Applied Logic

2016

Abstract. We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements.Theorem 0.1 (Superstability from categoricity). Let K be a (< κ)-tame AEC with amalgamation. If κ = κ > LS(K) and K is categorical in a λ > κ, then:• K is stable in any cardinal µ with µ ≥ κ.• K is categorical in κ.• There is a type-full good λ-frame with underlying class K λ .Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length).Theorem 0.2 (A global independence notion from categoricity). Let K be a densely type-local, fully tame and type short AEC with amalgamation. If K is categorical in unboundedly many cardinals, then there exists λ ≥ LS(K) such that K ≥λ admits a global independence relation with the properties of forking in a superstable first-order theory.As an application, we deduce (modulo an unproven claim of Shelah) that Shelah's eventual categoricity conjecture for AECs (without assuming categoricity in a successor cardinal) follows from the weak generalized continuum hypothesis and a large cardinal axiom.Corollary 0.3. Assume 2 λ < 2 λ + for all cardinals λ, as well as an unpublished claim of Shelah. If there exists a proper class of strongly compact cardinals, then any AEC categorical in some high-enough cardinal is categorical in all high-enough cardinals.

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

Mathematical Logic Qtrly

2018

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).