arXiv subjects
Publications and source records attributed to Christian Espíndola.
We give a detailed and self-contained introduction to the theory of $λ$-toposes and prove the following: 1) A $λ$-separable $λ$-topos has enough $λ$-points. 2) The classifying $λ$-topos of a $κ$-site $(\mathcal{C},E)$ is a presheaf topos (assuming $κ\vartriangleleft λ=λ^{<λ}$, $|\mathcal{C}|,|E|<λ$).
We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \geq μ$). When considering cardinalities of models of infinitary theories $\mathbb{T}$ of $\mathcal{L}_{κ, θ}$ that axiomatize $\mathcal{K}$, the result implies, under $SCH$, the following infinitary version of Morley's categoricity theorem: let $S$ be the class of cardinals $λ$ which are of cofinality at least $θ$ but are not successors of cardinals of cofinality less than $θ$. Then, if $\mathbb{T}$ is a $\mathcal{L}_{κ, θ}$ theory whose models have directed colimits and it is $λ$-categorical for some $λ\geq μ$ in $S$, then it is $λ'$-categorical for every $λ' \geq μ$ in $S$; moreover, we also exhibit an example that shows that the exceptions in the class $S$ are needed. Along the way we also prove Grossberg conjecture, according to which categoricity in a high enough cardinal implies eventual amalgamation. We establish this result in AEC's and, assuming in addition $SCH$, in the more general context of accessible categories whose morphisms are monomorphisms.
Given a weakly compact cardinal $κ$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel for intuitionistic logic over $\mathcal{L}_{ω_1, ω}$. When $κ$ is a regular cardinal such that $κ^{<κ}=κ$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over $\mathcal{L}_{κ^+, κ, κ}$, the language with disjunctions of at most $κ$ formulas, conjunctions of less than $κ$ formulas and quantification on less than $κ$ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.
We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized Continuum Hypothesis ($GCH$), for abstract elementary classes (AEC's) with interpolation, a strengthening of amalgamation which is a necessary and sufficient condition for an AEC categorical in a high enough cardinal to satisfy eventual categoricity. The proof builds on an earlier topos-theoretic argument which was syntactic in nature and recurred to $κ$-classifying toposes. We carry out here the same proof idea but from the semantic perspective, making use of a connection between $κ$-classifying toposes on one hand and the Scott adjunction on the other hand, this latter developed independently.
We prove preservation theorems for $\mathcal{L}_{ω_1, G}$, the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of Łoś-Tarski (resp. Lyndon) on sentences of $\mathcal{L}_{ω_1, ω}$ preserved by substructures (resp. homomorphic images). The solution, in $ZFC$, only uses general features and can be extended to several variants of other strong first-order logic that do not satisfy the interpolation theorem; instead, the results on infinitary definability are used. This solves an open problem dating back to 1977. Another consequence of our approach is the equivalence of the Vopěnka principle and a general definability theorem on subsets preserved by homomorphisms.
Given a regular cardinal $κ$ such that $κ^{<κ}=κ$ (e.g., if the Generalized Continuum Hypothesis holds), we develop a proof system for classical infinitary logic that includes heterogeneous quantification (i.e., infinite alternate sequences of quantifiers) within the language $\mathcal{L}_{κ^+, κ}$, where there are conjunctions and disjunctions of at most $κ$ any formulas and quantification (including the heterogeneous one) is applied to less than $κ$ many variables. This type of quantification is interpreted in $\mathcal{S}et$ using the usual second-order formulation in terms of strategies for games, and the axioms are based on a stronger variant of the axiom of determinacy for game semantics. Within this axiom system we prove the soundness and completeness theorem with respect to a class of set-valued structures that we call well-determined. Although this class is more restrictive than the class of determined structures in Takeuti's determinate logic, the completeness theorem works in our case for a wider variety of formulas of $\mathcal{L}_{κ^+, κ}$, and the category of well-determined models of heterogeneous theories is accessible. Our system is formulated within the sequent style of categorical logic and we do not need to impose any specific requirements on the proof trees, disregarding thus the eigenvariable conditions needed in Takeuti's system. We also investigate intuitionistic systems with heterogeneous quantifiers and prove analogously a completeness theorem with respect to well-determined structures in categories in general, in $κ$-Grothendieck toposes in particular, and, when $κ^{<κ}=κ$, also in Kripke models. Finally, we consider an extension of our system in which heterogeneous quantification with bounded quantifiers is expressible, and extend our completeness results to that case.
We give an analysis and generalizations of some long-established constructive completeness results in terms of categorical logic and pre-sheaf and sheaf semantics. The purpose is in no small part conceptual and organizational: from a few basic ingredients arises a more unified picture connecting constructive completeness with respect to Tarski semantics, to the extent that it is available, with various completeness theorems in terms of presheaf and sheaf semantics (and thus with Kripke and Beth semantics). From this picture are obtained both ("reverse mathematical") equivalence results and new constructive completeness theorems; in particular, the basic set-up is flexible enough to obtain strong constructive completeness results for languages of arbitrary size and languages for which equality between the elements of the signature is not decidable.
Given a regular cardinal $κ$ such that $κ^{<κ}=κ$, we study a class of toposes with enough points, the $κ$-separable toposes. These are equivalent to sheaf toposes over a site with $κ$-small limits that has at most $κ$ many objects and morphisms, the (basis for the) topology being generated by at most $κ$ many covering families, and that satisfy a further exactness property $T$. We prove that these toposes have enough $κ$-points, that is, points whose inverse image preserve all $κ$-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $κ=ω$, when property $T$ is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $κ$-geometric, where conjunctions of less than $κ$ formulas and existential quantification on less than $κ$ many variables is allowed. We prove that $κ$-geometric theories have a $κ$-classifying topos having property $T$, the universal property being that models of the theory in a Grothendieck topos with property $T$ correspond to $κ$-geometric morphisms (geometric morphisms the inverse image of which preserves all $κ$-small limits) into that topos. Moreover, we prove that $κ$-separable toposes occur as the $κ$-classifying toposes of $κ$-geometric theories of at most $κ$ many axioms in canonical form, and that every such $κ$-classifying topos is $κ$-separable. Finally, we consider the case when $κ$ is weakly compact and study the $κ$-classifying topos of a $κ$-coherent theory (with at most $κ$ many axioms), that is, a theory where only disjunction of less than $κ$ formulas are allowed, obtaining a version of Deligne's theorem for $κ$-coherent toposes.
We present a unified categorical treatment of completeness theorems for several classical and intuitionistic infinitary logics with a proposed axiomatization. This provides new completeness theorems and subsumes previous ones by Gödel, Kripke, Beth, Karp, Joyal, Makkai and Fourman/Grayson. As an application we prove, using large cardinals assumptions, the disjunction and existence properties for infinitary intuitionistic first-order logics.