We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a single \({\mathcal {L}}_{\omega _1,\omega }\)-sentence \(\psi \) that codes Kurepa trees to prove the following statements:

1. (1)

   The spectrum of \(\psi \) is consistently equal to \([\aleph _0,\aleph _{\omega _1}]\) and also consistently equal to \([\aleph _0,2^{\aleph _1})\), where \(2^{\aleph _1}\) is weakly inaccessible.

2. (2)

   The amalgamation spectrum of \(\psi \) is consistently equal to \([\aleph _1,\aleph _{\omega _1}]\) and \([\aleph _1,2^{\aleph _1})\), where again \(2^{\aleph _1}\) is weakly inaccessible. This is the first example of an \({\mathcal {L}}_{\omega _1,\omega }\)-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in Souldatos (Notre Dame J Form Log 55(4):533–551, 2014).

3. (3)

   Consistently, \(\psi \) has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in Baldwin et al. (Arch Math Log 55(3):545–565, 2016) and Baldwin and Souldatos (Math Log Q 65(4):444–452, 2019) of sentences with maximal models in countably many cardinalities.

4. (4)

   Consistently, \(2^{\aleph _0}<\aleph _{\omega _1}<2^{\aleph _1}\) and there exists an \({\mathcal {L}}_{\omega _1,\omega }\)-sentence with models in \(\aleph _{\omega _1}\), but no models in \(2^{\aleph _1}\). This relates to a conjecture by Shelah that if \(\aleph _{\omega _1}<2^{\aleph _0}\), then any \({\mathcal {L}}_{\omega _1,\omega }\)-sentence with a model of size \(\aleph _{\omega _1}\) also has a model of size \(2^{\aleph _0}\). Our result proves that \(2^{\aleph _0}\) can not be replaced by \(2^{\aleph _1}\), even if \(2^{\aleph _0}<\aleph _{\omega _1}\).

我们使用集合理论工具做出模型理论的贡献。特别是，我们构建了一个单一 \（{\ mathcal {L}} _ {\欧米加_1，\欧米加} \） -sentence \（\ PSI \）该代码Kurepa树木证明以下语句：

1. （1）

   \（\ psi \）的频谱始终等于\（[\ aleph _0，\ aleph _ {\ omega _1}] \），并且也始终等于\（[\ aleph _0,2 ^ {\ aleph _1} ）\），其中\（2 ^ {\ aleph _1} \）难以访问。

2. （2）

   \（\ psi \）的合并频谱始终等于\（[\ aleph _1，\ aleph _ {\ omega _1}] \）和\（[\ aleph _1,2 ^ {\ aleph _1}）\），再次很难访问\（2 ^ {\ aleph _1} \）。这是\（{\ mathcal {L}} _ {\ omega _1，\ omega} \）句子的第一个示例，其频谱和合并频谱始终右开和右闭。它还为Souldatos中的一个问题提供了肯定的答案（Notre Dame J Form Log 55（4）：533-551，2014）。

3. （3）

   一致地，\（\ psi \）具有有限，可数和不可数的许多基数的最大模型。这补充了鲍德温等人给出的例子。（Arch Math Log 55（3）：545-565，2016）和Baldwin and Souldatos（Math Log Q 65（4）：444-452，2019）的句子具有最大数量的基数的句子。

4. （4）

   一贯地，\（2 ^ {\ aleph _0} <\ aleph _ {\ omega _1} <2 ^ {\ aleph _1} \）并存在一个\（{\ mathcal {L}} _ {\ omega _1，\ omega} \）-使用\（\ aleph _ {\ omega _1} \）中的模型来句子，但是\（2 ^ {\ aleph _1} \）中没有模型。这与谢拉的猜想有关，如果\（\ aleph _ {\ omega _1} <2 ^ {\ aleph _0} \），则任何\（{\ mathcal {L}} _ {\ omega _1，\ omega} \）-具有大小\（\ aleph _ {\ omega _1} \）的模型的句子也具有大小\（2 ^ {\ aleph _0} \）的模型。我们的结果证明\（2 ^ {\ aleph _0} \）不能用\（2 ^ {\ aleph _1} \）代替，即使\（2 ^ {\ aleph _0} <\ aleph _ {\ omega _1} \）。