arXiv subjects

Will Boney

Publications and source records attributed to Will Boney.

At least 19 recordsLinked to original sources

arXiv2025

Model-theoretic characterizations of large cardinals (Re)${}^2$visited

We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $\Pi_n$-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vop\v{e}nka's Principle, are characterized by compactness properties involving Henkin models for sort logic. This provides a model-theoretic analogy between Vop\v{e}nka's Principle and weak Vop\v{e}nka's Principle. We also characterize huge cardinals by compactness for type omission properties of the well-foundedness logic $\mathbb L(Q^{\text{WF}})$, and show that the compactness number of the H\"artig quantifier logic $\mathbb L(I)$ can consistently be larger than the first supercompact cardinal. Finally, we show that the upward L\"owenheim-Skolem-Tarski number of second-order logic $\mathbb L^2$ and the sort logic $\mathbb L^{s,n}$ are given by the first extendible and $C^{(n)}$-extendible cardinal, respectively.

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arXiv2025

A Module-theoretic Introduction to Abstract Elementary Classes

The first-order model theory of modules has been studied for decades. More recently, the model theoretic study of nonelementary classes of modules--especially Abstract Elementary Classes of modules--has produced interesting results. This survey aims to discuss these recent results and give an introduction to the framework of Abstract Elementary Classes for module theorists.

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arXiv2024

$\Sigma_1$-Stationary logic as an $\aleph_1$-Abstract Elementary Class

$\mu$-Abstract Elementary Classes are a model theoretic framework introduced in [BGL+16] to encompass classes axiomatized by $\mathbb{L}_{\infty, \infty}$. We show that the framework extends beyond these logics by showing classes axiomatized in $\mathbb{L}(aa)$ with just the $aa$ quantifier are an $\aleph_1$-Abstract Elementary Class.

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arXiv2022

Cofinality quantifiers in Abstract Elementary Classes and beyond

The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemization that can prove a wide range of examples are Abstract Elementary Classes.

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arXiv2022

Model Theoretic Characterizations of Large Cardinals Revisited

In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.

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arXiv2022

A Lower Bound for the Hanf Number for Joint Embedding

In [13] the authors show that if $μ$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<μ$, and $K$ satisfies joint embedding (amalgamation) cofinally below $μ$, then $K$ satisfies joint embedding (amalgamation) in all cardinals $\ge μ$. The question was raised if the strongly compact upper bound was optimal. In this paper we prove the existence of an AEC $K$ that can be axiomatized by an $\mathcal{L}_{ω_1,ω}$-sentence in a countable vocabulary, so that if $μ$ is the first measurable cardinal, then (1) $K$ satisfies joint embedding cofinally below $μ$ ; (2) $K$ fails joint embedding cofinally below $μ$; and (3) $K$ satisfies joint embedding above $μ$. Moreover, the example can be generalized to an AEC $K^χ$ axiomatized in $\mathcal{L}_{χ^+, ω}$, in a vocabulary of size $χ$, such that (1)-(3) hold with $μ$ being the first measurable above $χ$. This proves that the Hanf number for joint embedding is contained in the interval between the first measurable and the first strongly compact. Since these two cardinals can consistently coincide, the upper bound from [13] is consistently optimal. This is also the first example of a sentence whose joint embedding spectrum is (consistently) neither an initial nor an eventual interval of cardinals. By Theorem 3.26, it is consistent that for any club $C$ on the first measurable $μ$, JEP holds exactly on $\lim C$ and everywhere above $μ$.

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arXiv2020

Which Classes of Structures Are Both Pseudo-elementary and Definable by an Infinitary Sentence?

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $\mathcal{L}_{ω_1 ω}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.

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arXiv2020

Tameness, powerful images, and large cardinals

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [BU17], [BTR16], [Lie18], and [LR16].

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arXiv2019

Categoricity in multiuniversal classes

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a result of the second author.

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arXiv2019

Model-theoretic Characterizations of Large Cardinals

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{κ, κ}$, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle.

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arXiv2018

Structural Logic and Abstract Elementary Classes with Intersection

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 AECs with intersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski's syntactic characterization of universal classes. As a corollary, we obtain that any AEC with countable Löwenheim-Skolem number is axiomatizable in $\mathbb{L}_{\infty, ω} (Q)$, where $Q$ is the quantifier "there exists uncountably many".

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arXiv2018

Erd\H{o}s-Rado Classes

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erd\H{o}s-Rado Theorem--into the notion of a combinatorial Erd\H{o}s-Rado class. These classes are closely related to Erd\H{o}s-Rado classes, which are those from which we can build generalized indiscernibles and blueprints in nonelementary classes, especially Abstract Elementary Classes. We give several examples and some applications.

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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

The Γ-Ultraproduct and Averageable Classes

We consider an ultraproduct that is designed to omit a fixed set of unary types $Γ$, called the $Γ$-ultraproduct. The $Γ$-ultraproduct is not always well-behaved, but we discuss several general conditions under which it is and several examples.

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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.

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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

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.

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arXiv2017

Forking in Short and Tame Abstract Elementary Classes

We develop a notion of forking for Galois-types in the context of Abstract Elementary Classes (AECs). Under the hypotheses that an AEC $K$ is tame, type-short, and failure of an order-property, we consider {\bf Definition.} Let $M_0 \prec N$ be models from $K$ and $A$ be a set. We say that the Galois-type of $A$ over $M$ \emph{does not fork over $M_0$} iff for all small $a \in A$ and all small $N^- \prec N$, we have that Galois-type of $a$ over $N^-$ is realized in $M_0$. Assuming property (E) (see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a \big cardinal". Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [BGKV] it is established that this notion of non-forking is the only independence relation possible.

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