The joint embedding property and maximal models
The joint embedding property and maximal models
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联合嵌入属性和最大模型
DOI：
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发表时间：
2015
影响因子：
0.3
通讯作者：
I. Souldatos
中科院分区：
文献类型：
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作者：
J. Baldwin;Martin Koerwien;I. Souldatos
我们引入了“纯”抽象基本类的概念来阻止琐碎的反例。我们研究了二部图的几类模型，并证明了：主要定理(参见。定理3.34和推论3.38)：如果$$Langellambda_i：Ileα<；Aleph_1
角度$$⟨λI：I≤α<；ℵ1⟩是模型满足jep$$(<；lambda_0)$$(<；λ0)的可刻画基数的严格递增序列(定义为2.1)，有一个$$L_{omega_1，omega}$$Lω1，ω-语句$$psi$$ψ，其模型形成一个纯AEC；(1)$$psi$$ψ的模型满足jep$$(<；lambda_0)$(<；λ0)，而JEP对于所有较大的基数都不成立，而AP在所有无限个基数中都不成立。(2)在$$λ_i^+$$ψ中存在$$psi$$λi+的非同构极大模型，对于所有$$ileα$$i≤α，但在任何其他基数中没有极大模型；以及(3)$$psi$$ψ具有任意大的模型。特别地，这给出了JEP的Hanf数和具有LöWENHEIM数$$Aleph_0$$ℵ0的纯AEC的最大化Hanf数至少是$$eth_{omega_1}$$ℶω1。我们证明了尽管每个$$kappa$$κ的$$ap(Kappa)$$ap(κ)蕴含着完全融合性质，但每个$$kappa$$κ的$$jep(Kappa)$$jep(κ)并不意味着完全联合嵌入性质。我们证明了本文的主要组合手段不能用来将主要定理推广到一个完整的句子。
We introduce the notion of a ‘pure’ Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.34 and Corollary 3.38): If $$langle lambda _i: ile alpha <aleph _1
angle $$⟨λi:i≤α<ℵ1⟩ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$$(<lambda _0)$$(<λ0), there is an $$L_{omega _1,omega }$$Lω1,ω-sentence $$psi $$ψ whose models form a pure AEC and(1)The models of $$psi $$ψ satisfy JEP$$(<lambda _0)$$(<λ0), while JEP fails for all larger cardinals and AP fails in all infinite cardinals.(2)There exist $$2^{lambda _i^+}$$2λi+ non-isomorphic maximal models of $$psi $$ψ in $$lambda _i^+$$λi+, for all $$ile alpha $$i≤α, but no maximal models in any other cardinality; and(3)$$psi $$ψ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Löwenheim number $$aleph _0$$ℵ0 are at least $$eth _{omega _1}$$ℶω1. We show that although $$AP(kappa )$$AP(κ) for each $$kappa $$κ implies the full amalgamation property, $$JEP(kappa )$$JEP(κ) for each $$kappa $$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.