Papers by christian espindola
Boletín de la Sociedad Chilena de Arqueología, Dec 30, 2023
Dossier: Arqueología, patrimonio, archivos y museos 27-31. Arqueología, patrimonio, archivos y mu... more Dossier: Arqueología, patrimonio, archivos y museos 27-31. Arqueología, patrimonio, archivos y museos. Presentación Leonor Adán 32-52. El redescubrimiento de la arqueología de las tierras bajas bolivianas en los repositorios del Museo de La Plata. Algunos resultados del proyecto SciCoMove Irina Podgorny, Nathalie Richard y Carla Jaimes Betancourt 53-79. Desafíos y amenazas del MAPSE Museo Rapa Nui en el contexto de descolonización: entre la adaptación y la incertidumbre María Gabriela Atallah Leiva 80-93.
arXiv (Cornell University), Jun 21, 2019
We provide a proof, in ZF C, of Shelah's eventual categoricity conjecture for abstract elementary... more We provide a proof, in ZF C, 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 K is such a category, we show that there is a cardinal µ such that if K is λ-categorical for some λ ≥ µ (i.e., it has only one object of internal size λ up to isomorphism), then K is eventually categorical (i.e., it is λ ′-categorical for every λ ′ ≥ µ). When considering cardinalities of models of infinitary theories T of L κ,θ that axiomatize 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 T is a L κ,θ theory whose models have directed colimits and it is λ-categorical for some λ ≥ µ in S, then it is λ ′-categorical for every λ ′ ≥ 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. 1
Journal of Symbolic Logic, Sep 1, 2020
Given a regular cardinal κ such that κ <κ = κ (or any regular κ if the Generalized Continuum Hypo... more Given a regular cardinal κ such that κ <κ = κ (or any regular κ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the κ-separable toposes. These are equivalent to sheaf toposes over a site with κsmall 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 from which we can derive, among other things, Karp's completeness theorem for infinitary classical logic.
arXiv (Cornell University), Sep 30, 2019
We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized C... more 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. 1 1 This research has been supported through the grant 19-00902S from the Grant Agency of the Czech Republic. I would like to thank Jiří Rosický and Ivan Di Liberti for reading and commenting a first version.
arXiv (Cornell University), Jun 21, 2019
We provide a proof, in ZF C, of Shelah's eventual categoricity conjecture for abstract elementary... more We provide a proof, in ZF C, 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 K is such a category, we show that there is a cardinal µ such that if K is λ-categorical for some λ ≥ µ (i.e., it has only one object of internal size λ up to isomorphism), then K is eventually categorical (i.e., it is λ ′-categorical for every λ ′ ≥ µ). When considering cardinalities of models of infinitary theories T of L κ,θ that axiomatize 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 T is a L κ,θ theory whose models have directed colimits and it is λ-categorical for some λ ≥ µ in S, then it is λ ′-categorical for every λ ′ ≥ 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. 1
arXiv (Cornell University), Sep 30, 2019
We provide a short proof of Shelah's eventual categoricity conjecture, assuming the Generalized C... more 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. 1 1 This research has been supported through the grant 19-00902S from the Grant Agency of the Czech Republic. I would like to thank Jiří Rosický and Ivan Di Liberti for reading and commenting a first version.
arXiv (Cornell University), Jun 21, 2019
We prove preservation theorems for L ω1,G , the countable fragment of Vaught's closed game logic.... more We prove preservation theorems for L ω1,G , the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of Loś-Tarski (resp. Lyndon) on sentences of L ω1,ω preserved by substructures (resp. homomorphic images). The solution, in ZF C, 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.
arXiv (Cornell University), Jan 31, 2019
Given a regular cardinal κ such that κ <κ = κ (e.g. if the Generalized Continuum Hypothesis holds... more 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 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 Set 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 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 for L κ + ,κ,κ (when only conjunctions of less than κ many formulas are allowed), 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.
arXiv (Cornell University), Jun 15, 2018
Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic ... more Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over 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 in [Nad78] for intuitionistic logic over L ω1,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 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.
arXiv (Cornell University), Jun 21, 2019
We provide a proof, in ZF C, of Shelah's eventual categoricity conjecture for abstract elementary... more We provide a proof, in ZF C, 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 K is such a category, we show that there is a cardinal µ such that if K is λ-categorical for some λ ≥ µ (i.e., it has only one object of internal size λ up to isomorphism), then K is eventually categorical (i.e., it is λ ′-categorical for every λ ′ ≥ µ). When considering cardinalities of models of infinitary theories T of L κ,θ that axiomatize 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 T is a L κ,θ theory whose models have directed colimits and it is λ-categorical for some λ ≥ µ in S, then it is λ ′-categorical for every λ ′ ≥ 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. 1
arXiv (Cornell University), Jan 5, 2017
We present a unified categorical treatment of completeness theorems for several classical and int... more 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.
This thesis is an exploration of several completeness phenomena, both in the constructive and the... more This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems. The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold. Sammanfattning Denna avhandling är en undersökning av flera fullständighetsfenomen, både i konstruktiva och klassiska versioner. Efter några inledande kapitel i den första delen av avhandlingen där vi beskriver bakgrunden som kommer att användas senare, innehåller den konstruktiva delen en kategoriteoretisk formulering av flera konstruktiva fullständighetssatser som finns i litteraturen, men här presenterade i ett enhetligt ramverk. Vi utvecklar dem från den konstruktiva "omvända matematikens" perspektiv, med fokus på metateorien som används i varje enskilt fall och styrkan hos motsvarande fullständighetssatser. Den klassiska delen av avhandlingen fokuserar på infinitär intuitionistisk satslogik och predikatlogik. Vi betraktar ett axiomatiskt satslogiskt system med en speciell distributivitetsregel som är tillräcklig för att bevisa en fullständighetssats, och vi introducerar svagt kompakta kardinaltal som det adekvata metateoretiska antagande. Slutligen återvänder vi till den kategoriska formuleringen, fokuserande denna gång på infinitär första ordningens intuitionistisk logik. Vi föreslår ett predikatssystem med en särskild regel, transfinit transitivitet, som innehåller både distributivitet och en form av axiomet om beroende urval, och studerar i vilken utsträckning fullständighetssatser gäller. Vi fastlägger fullständighet under antagande om ett svagt kompakt kardinaltal, och studerar det disjunktionsfria fragmentet precis som i avhandlingens konstruktiva del. Antagandet om svag kompakthet bevisas vara väsentligt för att fullständighetssatserna ska gälla.
Frontiers in Earth Science
Since ancient times Andean societies have formed an intimate relationship with volcanoes, the beg... more Since ancient times Andean societies have formed an intimate relationship with volcanoes, the beginnings of which can be traced right back to the initial peopling of the region. By studying rocks used for stone tools and other everyday artifacts, we explore the volcanic landscapes of early hunter-gatherer groups (11,500–9,500 cal BP) of the highlands of the Atacama Desert (22–24°S/67–68°W). Petrological classification of the lithic assemblages of three Early Holocene archaeological sites showed the procurement of a great diversity of volcanic and subvolcanic rocks, including pumice, granitic rocks, micro-diorites, a large variety of tuffs and andesites, dacites, cherts, basalts, obsidians, among others. Field surveys enabled us to detect many of their sources related to volcanic features such as craters, maars, caldera-domes, lava flows, probable hydrothermal deposits, and ignimbrites. In these places, we also document large quarry-workshops and campsites from different periods, ind...
This thesis is an exploration of several completeness phenomena, both in the constructive and the... more This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outli ...
The Journal of Symbolic Logic, 2020
Notre Dame Journal of Formal Logic, 2016
We introduce a general notion of semantic structure for first-order theories, covering a variety ... more We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean Prime Ideal theorem (BPI). In particular, we deduce that the completeness of that type of semantics for non-classical theories is unprovable in intuitionistic Zermelo Fraenkel set theory IZF. Using results of Joyal and McCarty, we conclude also that the completeness of Kripke semantics is equivalent, over IZF, to the Law of Excluded Middle plus BPI. By results from Banaschewski and Bhutani, none of these two principles imply each other, and so this gives the exact strength of Kripke completeness theorem in the sense of constructive reverse mathematics.
Techné/Technè, Dec 31, 2023
arXiv (Cornell University), Sep 18, 2017
We give an analysis and generalizations of some long-established constructive completeness result... more We give an analysis and generalizations of some long-established constructive completeness results in terms of categorical logic and presheaf 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 setup 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. MSC2010 classification: 03F99, 03G30.
Diálogo andino
Presentamos los resultados de una investigación sobre formas históricas de movilidad y ocupación ... more Presentamos los resultados de una investigación sobre formas históricas de movilidad y ocupación territorial-en el desierto y la puna de Atacama-habilitadas por el ciclo de pastoreo en combinación con viajes de comercio o arrierías intercomunitarias. En primer lugar, explicamos por qué caracterizamos a esta investigación como colaborativa. En segundo lugar, damos cuenta de la metodología implementada (articulación del trabajo, fuentes de información, complementación). En tercer lugar, brindamos un paneo de los resultados obtenidos. La relevancia se centra tanto en los aportes al estudio sobre las arrierías de complementación entre comunidades andinas, como en demostrar las posibles articulaciones entre los objetivos de las comunidades y los de la investigación antropológica.
arXiv (Cornell University), Jan 30, 2023
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Papers by christian espindola