Infinitary generalizations of Deligne's completeness theorem

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F20%3A00118489" target="_blank" >RIV/00216224:14310/20:00118489 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1017/jsl.2020.27" target="_blank" >https://doi.org/10.1017/jsl.2020.27</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/jsl.2020.27" target="_blank" >10.1017/jsl.2020.27</a>

Result language

angličtina

Original language name

Infinitary generalizations of Deligne's completeness theorem

Original language description

Given a regular cardinal kappa such that kappa(&lt;kappa) = kappa (or any regular if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the kappa-separable toposes. These are equivalent to sheaf toposes over a site with kappa-small limits that has at most kappa many objects and morphisms, the (basis for the) topology being generated by at most. many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough kappa-points, that is, points whose inverse image preserve all kappa-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when kappa = omega, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call kappa-geometric, where conjunctions of less than. formulas and existential quantification on less than. many variables is allowed. We prove that kappa-geometric theories have kappa-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to kappa-geometric morphisms (geometric morphisms the inverse image of which preserves all kappa-small limits) into that topos. Moreover, we prove that kappa-separable toposes occur as the kappa-classifying toposes of kappa-geometric theories of at most. many axioms in canonical form, and that every such kappa-classifying topos is kappa-separable. Finally, we consider the case when. is weakly compact and study the kappa-classifying topos of a kappa-coherent theory (with at most. many axioms), that is, a theory where only disjunction of less than. formulas are allowed, obtaining a version of Deligne's theorem for.-coherent toposes from which we can derive, among other things, Karp's completeness theorem for infinitary classical logic.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2020

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Symbolic Logic

ISSN

0022-4812

e-ISSN

—

Volume of the periodical

85

Issue of the periodical within the volume

3

Country of publishing house

GB - UNITED KINGDOM

Number of pages

16

Pages from-to

1147-1162

UT code for WoS article

000628900500012

EID of the result in the Scopus database

2-s2.0-85102715919