We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint$\left( { \le \lambda ,k} \right)$-amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.
Abstract. We introduce the notion of a 'pure' Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show:Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If λ i : i ≤ α < ℵ 1 is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP(< λ 0 ), there is an Lω 1 ,ω -sentence ψ whose models form a pure AEC and(1) The models of ψ satisfy JEP(< λ 0 ), while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist 2 λ + i non-isomorphic maximal models of ψ in λ + i , for all i ≤ α, but no maximal models in any other cardinality; and (3) ψ has arbitrarily large models.In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Löwenheim number ℵ 0 are at least ω 1 . We show that although AP (κ) for each κ implies the full amalgamation property, JEP (κ) for each κ does not imply the full joint embedding property.We prove the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.We investigate in this paper the spectra of joint embedding and of maximal models for an Abstract Elementary Class (AEC), in particular for AEC defined by universal L ω1,ω -sentences under substructure. Our main result provides a collection of bipartite graphs whose combinatorics allows us to construct for any given countable strictly increasing sequence of characterizable cardinals (λ i ), a sentence of L ω1,ω whose models have joint embedding below λ 0 and 2 λi + -many maximal models in each λ + i , but arbitrarily large models. Two examples of such sequences (λ i ) are: (1) an enumeration of an arbitrary countable subset of the α , α < ω 1 , and (2) an enumeration of an arbitrary countable subset of the ℵ n , n < ω.We give precise definitions and more details in Section 1. In Section 2, we describe our basic combinatorics and the main constructions are in Section 3. We now provide some background explaining several motivations for this study.In first order logic, work from the 1950's deduces syntactic characterizations of such properties as joint embedding and amalgamation via the compactness theorem. The syntactic conditions immediately yield that if these properties hold in one cardinality they hold in all cardinalities. For AEC this situation is vastly different. In fact, one major stream studies what are sometimes called Jónsson classes that satisfy: amalgamation, joint embedding, and have arbitrarily large models. (See, for example, [?, ?, ?] and a series of paper such as [?].) Without this hypothesis the properties must be parameterized and the relationship between, e.g. the Joint Embedding Property (JEP) holding in various cardinals, becomes a topic for study. In [?] Grossberg conjectures the existence of a Hanf number for the Amalgamation Property (AP): a cardinal µ(λ) such that if an AEC with Löwenheim number λ has the AP in some cardinal greater than µ(λ) then it has the amalgamation property in all lar...
We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.
The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. The fundamental theorem on ordinal computability states that a set x of ordinals is ordinal computable from ordinal parameters if and only if x is an element of the constructible universe L.I nt h i s paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo-Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.