arXiv subjects

Sebastien Vasey

Publications and source records attributed to Sebastien Vasey.

At least 37 records · Page 2Linked to original sources

arXiv2018

Abstract elementary classes stable in $\aleph_0$

We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at $\aleph_0$. More precisely, there is a superlimit model of cardinality $\aleph_0$ and the class generated by this superlimit has a type-full good $\aleph_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality $\aleph_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.

math.LO↗

arXiv2017

Good Frames in the Hart-Shelah Example

For a fixed natural number $n \geq 1$, the Hart-Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to $\aleph_n$. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good $\aleph_{n-1}$-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the way, we develop new tools to build and analyze good frames.

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arXiv2017

Quasiminimal abstract elementary classes

We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim-Skolem-Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber's quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber's definition of a quasiminimal pregeometry class.

math.LO↗

arXiv2017

On the uniqueness property of forking in abstract elementary classes

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.

math.LO↗

arXiv2017

Saturation and solvability in abstract elementary classes with amalgamation

$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $λ> \text{LS} (K)$. 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): $\mathbf{Corollary.}$ Let $K$ be an AEC with amalgamation and no maximal models. Let $λ> μ> \text{LS} (K)$. If $K$ is solvable in $λ$, then $K$ is solvable in $μ$.

math.LO↗

arXiv2017

Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: $\mathbf{Theorem}$ If $K$ is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $λ$: * The union of an increasing chain of $λ$-saturated models is $λ$-saturated. * There exists a type-full good $λ$-frame with underlying class the saturated models of size $λ$. * There exists a unique limit model of size $λ$. Our proofs use independence calculus and a generalization of averages to this non first-order context.

math.LO↗

arXiv2017

Shelah-Villaveces revisited

We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $μ^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $μ^{+}, μ^{++}, \ldots, μ^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of cardinals imply categoricity on a tail of cardinals. This is the first categoricity transfer theorem in that setup and answers a question of Baldwin.

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arXiv2017

Shelah's eventual categoricity conjecture in universal classes: part I

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Assume that $K$ is fully $\operatorname{LS} (K)$-tame and short and has primes over sets of the form $M \cup \{a\}$. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\operatorname{LS} (K)}\right)^+}}\right)^+}$. If $K$ is categorical in a $λ> H_2$, then $K$ is categorical in all $λ' \ge H_2$.

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arXiv2017

Symmetry in abstract elementary classes with amalgamation

This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolated by the first author) in AECs with amalgamation that satisfy a local definition of superstability. The key results are a downward transfer of symmetry and a deduction of symmetry from failure of the order property. These results are then used to prove several structural properties in categorical AECs, improving classical results of Shelah who focused on the special case of categoricity in a successor cardinal. We also study the interaction of symmetry with tameness, a locality property for Galois (orbital) types. We show that superstability and tameness together imply symmetry. This sharpens previous work of Boney and the second author.

math.LO↗

arXiv2017

Equivalent definitions of superstability in tame abstract elementary classes

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent: $\mathbf{Corollary}$ Let $K$ be a tame AEC with a monster model. Assume that $K$ is stable in a proper class of cardinals. The following are equivalent: 1) For all high-enough $λ$, $K$ has no long splitting chains. 2) For all high-enough $λ$, there exists a good $λ$-frame on a skeleton of $K_λ$. 3) For all high-enough $λ$, $K$ has a unique limit model of cardinality $λ$. 4) For all high-enough $λ$, $K$ has a superlimit model of cardinality $λ$. 5) For all high-enough $λ$, the union of any increasing chain of $λ$-saturated models is $λ$-saturated. 6) There exists $μ$ such that for all high-enough $λ$, $K$ is $(λ, μ)$-solvable. This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

math.LO↗

arXiv2017

Shelah's eventual categoricity conjecture in tame AECs with primes

A new case of Shelah's eventual categoricity conjecture is established: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. Assume that $K$ is $H_2$-tame and $K_{\ge H_2}$ has primes over sets of the form $M \cup \{a\}$. If $K$ is categorical in some $λ> H_2$, then $K$ is categorical in all $λ' \ge H_2$. 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 theorem is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): $\mathbf{Theorem}$ Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is categorical in a $λ> |T|$, then $D$ is categorical in all $λ' \ge \min (λ, \beth_{(2^{|T|})^+})$.

math.LO↗

arXiv2017

Superstability from categoricity in abstract elementary classes

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof.

math.LO↗

arXiv2017

Tameness and frames revisited

We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah's notion of dimension is well-behaved, complementing previous work of Jarden and Sitton. We also improve a result of the first author on extending a frame to larger models.

math.LO↗

arXiv2016

Shelah's eventual categoricity conjecture in universal classes. Part II

We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the "high-enough" threshold: $\mathbf{Theorem}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence. If $ψ$ is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then $ψ$ is categorical in all $λ' \ge \beth_{\beth_{ω_1}}$. As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes: $\mathbf{Corollary}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence that is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then the class of models of $ψ$ has the amalgamation property for models of size at least $\beth_{\beth_{ω_1}}$. We also establish generalizations of these two results to uncountable languages. As part of the argument, we develop machinery to transfer model-theoretic properties between two different classes satisfying a compatibility condition. This is used as a bridge between Shelah's milestone study of universal classes (which we use extensively) and a categoricity transfer theorem of the author for abstract elementary classes that have amalgamation, are tame, and have primes over sets of the form $M \cup \{a\}$.

math.LO↗

arXiv2016

Downward categoricity from a successor inside a good frame

We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: $\mathbf{Theorem}$ Let $K$ be an AEC and let $\text{LS} (K) \le λ< θ$ be cardinals. If $K$ has a type-full good $[λ, θ]$-frame and $K$ is categorical in both $λ$ and $θ^+$, then $K$ is categorical in all $λ' \in [λ, θ]$. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from $\beth_{\beth_{\left(2^{\text{LS} (K)}\right)^+}}$ to $\beth_{\left(2^{\text{LS} (K)}\right)^+}$ assuming that the AEC is $\text{LS} (K)$-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form $M \cup \{a\}$ or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.

math.LO↗

arXiv2016

A survey on tame abstract elementary classes

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract independence relations such as Shelah's good frames have been found to be key objects. Several new categoricity transfers have been obtained. We survey these developments using the following result (due to the second author) as our guiding thread: $\mathbf{Theorem}$ If a universal class is categorical in cardinals of arbitrarily high cofinality, then it is categorical on a tail of cardinals.

math.LO↗

arXiv2016

Building prime models in fully good abstract elementary classes

We show how to build primes models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: $\mathbf{Theorem.}$ Let $K$ be an almost fully good AEC that is categorical in $\text{LS} (K)$ and has the $\text{LS} (K)$-existence property for domination triples. For any $λ> \text{LS} (K)$, the class of Galois saturated models of $K$ of size $λ$ has prime models over every set of the form $M \cup \{a\}$. This generalizes an argument of Shelah, who proved the result when $λ$ is a successor cardinal.

math.LO↗

arXiv2016

Building independence relations in abstract elementary classes

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. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $(<κ)$-tame AEC with amalgamation. If $κ= \beth_κ> \text{LS} (K)$ and $K$ is categorical in a $λ> κ$, then: * $K$ is stable in all cardinals $\ge κ$. * $K$ 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). $\mathbf{Theorem}$ (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 $λ\ge \text{LS} (K)$ such that $K_{\ge λ}$ 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. $\textbf{Corollary}$ 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.

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