Abstract. This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence φ , in the sense that φ characterizes κ, if φ has a model of size κ, but no models of size κ + .We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if ℵα is characterized by a Scott sentence, at least one of ℵα and ℵ α+1 is homogeneously characterizable (cf. definition 1.3 and theorem 2.9). Based on Shelah's [8], we give counterexamples that characterizable cardinals are not closed under predecessors, or cofinalities.
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...
This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $\kappa$ is characterized by a Scott sentence $\phi_M$, if $\phi_M$ has a model of size $\kappa$, but no model of $\kappa^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_{\beta}$ is characterized by a Scott sentence, then $2^{\aleph_{\beta+\beta_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<\beta_1<\omega_1$. So, the answer to the above question is positive, except the case $\beta_1=0$ which remains open. As a consequence we derive that if $\alpha\le\beta$ and $\aleph_{\beta}$ is characterized by a Scott sentence, then $\aleph_{\alpha+\alpha_1}^{\aleph_{\beta+\beta_1}}$ is also characterized by a Scott sentence, for all $\alpha_1<\omega_1$ and $0<\beta_1<\omega_1$. Whence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are also mentioned at the end.Comment: This paper is an updated version of the second half of version 1 of arXiv:1007.2426v
In [BKS14] examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper we give examples of complete Lω 1 ,ωsentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of κ + , κ ω , 2 κ and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.We unite ideas from [BFKL13, BKL14, Hjo02, Kni77] to find complete sentences of L ω1,ω with maximal models in multiple cardinals. There have been a number of papers finding complete sentences characterizing cardinals beginning with Baumgartner, Malitz and Knight in the 70's, refined by Laskowski and Shelah in the 90's and crowned by Hjorth's characterization of all cardinals below ℵ ω1 in the 2002. These results have been refined since. But this is the first paper finding complete sentences with maximal models in two or more cardinals. All models of these sentences have cardinality less than ω1 .
The current paper answers an open question of [5]. We say that a countable model M characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ + . If M is linearly ordered by <, we will say that the linear ordering (M, <) characterizes κ, or that κ is characterizable by (M, <).From [2] we can deduce that if κ is characterizable, then κ + is characterizable by a linear ordering (see theorem 2.4, corollary 2.5). From [5] we know that if κ is characterizable by a dense linear ordering, then 2 κ is characterizable (see theorem 2.7).We show that if κ is homogeneously characterizable (cf. definition 2.2), then κ is characterizable by a dense linear ordering, while the converse fails (theorem 2.3).The main theorems are: 1) If κ > 2 λ is a characterizable cardinal, λ is characterizable by a dense linear ordering and λ is the least cardinal such that κ λ > κ, then κ λ is also characterizable (theorem 5.4), 2) if ℵα and κ ℵα are characterizable cardinals, then the same is true for κ ℵ α+β , for all countable β (theorem 5.5).Combining these two theorems we get that if κ > 2 ℵα is a characterizable cardinal, ℵα is characterizable by a dense linear ordering and ℵα is the least cardinal such that κ ℵα > κ, then for all β < α+ω 1 κ ℵ β is characterizable (theorem 5.7). Also if κ is a characterizable cardinal, then κ ℵα is characterizable, for all countable α (corollary 5.6). Structure of the paperThroughout the whole paper we work with countable languages L and when we refer to a dense linear ordering we mean a dense linear ordering without endpoints. The first two sections provide some background material for the characterizable cardinals and for the dense linear orderings respectively. Section 4 contains the construction that proves the following Theorem 1.1. If κ is a characterizable cardinal, then κ ℵ1 is also a characterizable cardinal.This appears as theorem 4.18 in section 4 and it will be easily generalized to λ ≥ ℵ 1 in the last section. Characterizable cardinalsThis section provides the necessary background on characterizable cardinals.
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