In BKSoul [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 Lω1,ωsentences with maximal models in more than one cardinality; indeed, consistently in countably many cardinalities. The key new construction is a complete Lω1,ω-sentence with arbitrarily large models but with (κ+, κ) models for every κ. We unite ideas from BFKL, BKL, Hjorthchar,Knightex [BFKL13, BKL14, Hjo02, Kni77] to find complete sentences with maximal models in two cardinals. There have been a number of papers finding complete sentences of Lω1,ω 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 אω1 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. Our arguments combine and extend the techniques of building atomic models by Fraissé constructions using disjoint amalgamation, pioneered by Laskowski-Shelah and Hjorth, with the notion of homogeneous characterization and tools from Baldwin-Koerwien-Laskowski. This paper combines the ideas of Hjorth and Knight with specific techniques from BFKL, BKL, SouldatosCharacterizableCardinals, Souldatoscharpow [BFKL13, BKL14, Sou14, Sou13] and many proofs are adapted from these sources. We thank the referee for a perceptive and helpful report. Structure of the paper: In Section template 1, we explain the merger techniques for combining sentences that homogeneously characterize one cardinal (possibly in terms of another) to get a single complete sentence with maximal models in prescribed cardinalities. Section genknight 2 contains the main technical construction of the paper: the existence of a complete sentence φ with a unary predicate that has (κ, κ) models for every κ. From this construction and the tools of Section template 1, we complete the proof of Theorem knightgen 1.5 and thus of Corollary Cor:kkplus 1.7: for each homogenously characterizable κ, examples of Lω1,ω-sentences φκ with maximal models in κ and κ and no larger models. In Section sec:ltoomega 3 we present examples, for each homogenously characterizable κ, of Lω1,ω-sentences with maximal models in κ and κ and no larger models. The argument can be generalized to maximal models in κ and κאα , for all countable α. Date: January 17, 2017. 2010 Mathematics Subject Classification. Primary 03C75, 03C35 Secondary 03C52, 03C30, 03C15.

中文翻译：

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用多基数的极大模型完成 Lω1,ω-句子

在 BKSoul [BKS14] 中，不完整句子的例子是用多个基数的极大模型给出的。提出了一个问题，是否可以找到完整句子的相似示例。在本文中，我们给出了具有多个基数的极大模型的完整 Lω1,ω 句子的示例；事实上，始终在可数的许多基数中。关键的新结构是一个完整的 Lω1,ω-句子，具有任意大的模型，但每个 κ 都有 (κ+, κ) 模型。我们将来自 BFKL、BKL、Hjorthchar、Knightex [BFKL13、BKL14、Hjo02、Kni77] 的想法结合起来，以在两个基数中找到具有最大模型的完整句子。有许多论文发现 Lω1,ω 的完整句子以 70 年代的 Baumgartner、Malitz 和 Knight 开头，表征红雀，90 年代由 Laskowski 和 Shelah 改进，并在 2002 年 Hjorth 对低于 אω1 的所有基数的表征加冕。此后这些结果已经改进。但这是第一篇在两个或更多基数中找到具有最大模型的完整句子的论文。我们的论点结合并扩展了使用不相交合并的 Fraissé 结构构建原子模型的技术，该技术由 Laskowski-Shelah 和 Hjorth 首创，同质表征的概念和来自 Baldwin-Koerwien-Laskowski 的工具。这篇论文将 Hjorth 和 Knight 的思想与来自 BFKL、BKL、SouldatosCharacterizableCardinals、Souldatoscharpow [BFKL13、BKL14、Sou14、Sou13] 的特定技术相结合，许多证明都是从这些来源改编而来的。我们感谢裁判的洞察力和有用的报告。论文结构：在模板1中，我们解释了合并技术，用于组合均匀表征一个基数（可能是另一个基数）的句子，以获得具有指定基数的最大模型的单个完整句子。genknight 2 部分包含论文的主要技术结构：存在一个完整的句子 φ，带有一个一元谓词，每个 κ 都有 (κ, κ) 模型。从这个构造和模板 1 的工具，我们完成了定理 Knightgen 1.5 和 Corollary Cor:kkplus 1.7 的证明：对于每个可同构的 κ，Lω1，ω-句子 φκ 的例子，在 κ 和 κ 中具有最大模型，并且没有更大的模型。在第 sec:ltoomega 3 节中，我们展示了 Lω1,ω 句子的每个可同构 κ 的示例，其中具有 κ 和 κ 中的最大模型，没有更大的模型。对于所有可数 α，该论证可以推广到 κ 和 κאα 中的极大模型。日期：2017 年 1 月 17 日。2010 年数学学科分类。初级 03C75、03C35 次​​级 03C52、03C30、03C15。