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Categoricity and amalgamation for AEC, and κ measurable
Categoricidad y amalgamación para clases elementales abstractas, y κ medible
DOI:
https://doi.org/10.15446/bol.mat.v31n1.122442Palabras clave:
Model theory, abstract elementary classes, AEC, categoricity, infinitary logic, amalgamation (en)teoría de modelos, clases elementales abstractas, AEC, categoricidad, lógica infinitaria, amalgamació (es)
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In the original version of this paper, we assume a theory T in the logic Lκ,ℵ0 is categorical in a cardinal λ > κ, and κ is a measurable cardinal. There we prove that the class of models of T of cardinality < λ but ≥ |T| + κ has the amalgamation property under a natural order; this is a step toward understanding the character of such classes of models.
In this revised version we replaced the class of models of T by k, an AEC (abstract elementary class) which has LST-number <κ, or at least which behaves nicely for ultra-powers by D, some normal ultra-filter on κ or just LST+k -complete non-principal ultra-filters on κ.
Presently sub-section §2A deals with T ⊆ Lκ+,ℵ0 (and so does a large part of the introduction and little in the rest of §2), but otherwise, all is done in the context of AEC.
The reader may in the first reading for transparency fix D, a normal ultrafilter on the measurable cardinal κ and either fix the T ⊆ Lκ,ℵ0 or fix an AEC k with LSTk < κ.
We leave the original introduction adding a few comments at the end, after the three stars.
En la versión original de este artículo, suponemos que una teoría T en la lógica Lκ,ℵ0 es categórica en un cardinal λ > κ, y que κ es un cardinal medible. Allí demostramos que la clase de modelos de T de cardinalidad < λ con ≥ |T|+κ tiene la propiedad de amalgamación bajo un orden natural; este es un paso hacia la comprensión del carácter de dichas clases de modelos.
En esta versión revisada, reemplazamos la clase de modelos de T por k, una AEC (clase elemental abstracta) que tiene número de Löwenheim-Skolem-Tarski < κ, o al menos que se comporta bien para ultrapotencias por D, algún ultrafiltro normal sobre κ o simplemente ultrafiltros no principales LST+k - completos sobre κ.
Actualmente, la subsección §2A trata con T ⊆ Lκ+,aleph0 (y también una gran parte de la introducción y poco en el resto del §2), pero por lo demás, todo se hace en el contexto de las AEC.
Para una primera lectura, y en aras de la transparencia, el lector puede fijar D un ultrafiltro normal sobre el cardinal medible κ, y bien fijar T ⊆ Lκ,ℵ0 o bien fijar una AEC k con LSTk < κ
Dejamos la introducción original añadiendo algunos comentarios al final, después de los tres asteriscos.
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