共尾性谱问题的模型论应用
Model-theoretic applications of cofinality spectrum problems
摘要：
我们应用最近发展起来的共尾性谱技术证明了模型论中的一系列定理。首先，我们证明了Peano算法的任何模型都是$λ$ -饱和的当且仅当它具有共尾性$\geq λ$，并且对于正则的$κ< λ$，其底层序没有$(κ, κ)$ -割.其次，假设GCH的实例，我们证明了$SOP_2$的最大性的可解释性顺序$\trianglelefteq^*$的特点，解决了一个先验猜想，并证明了$SOP_2$是一个真正的分界线。第三，我们建立了$NSOP_2$的结构理论的开端，证明了$NSOP_2$的特征在于存在少数不一致的高级公式。在本文中，我们证明了$\mathfrak{p}_s = \mathfrak{t}_s$在任何弱共尾性谱问题中在指数化（自然定义）下是闭的。我们还证明了这些基数的本地版本不需要一致，即使在共尾谱问题所产生的皮亚诺算术。
英文摘要：
We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is $λ$-saturated iff it has cofinality $\geq λ$ and the underlying order has no $(κ, κ)$-cuts for regular $κ< λ$. Second, assuming instances of GCH, we prove that $SOP_2$ characterizes maximality in the interpretability order $\trianglelefteq^*$, settling a prior conjecture and proving that $SOP_2$ is a real dividing line. Third, we establish the beginnings of a structure theory for $NSOP_2$, proving that $NSOP_2$ can be characterized by the existence of few inconsistent higher formulas. In the course of the paper, we show that $\mathfrak{p}_s = \mathfrak{t}_s$ in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.
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