Kaufmann模型的可破坏性和可公理化性
Destructibility and Axiomatizability of Kaufmann Models
摘要：
Kaufmann模型是一个类似$ω_1$的、递归饱和的、相当无类的$\mathrm{PA}$或$\mathrm{ZF}$模型. Kaufmann根据组合原理$\diamondsuit_{ω_1}$构造了这些模型，Shelah通过绝对性论证证明了它们存在于$\mathrm{ZFC}$中。Kaufmann模型是$ω_1$类似于Aronszajn树的非紧性的重要证明。在本文中，我们看一些集合论的问题与此相关的动机似乎是天真的问题，这样的模型是否可以“杀死”的强迫没有崩溃$ω_1$。我们表明，这个问题的答案是独立的$\mathrm{ZFC}$和密切相关的类似问题Aronszajn树。作为这些方法的一个应用，我们还证明了Kaufmann模型是否可以在逻辑$L_{ω_1, ω} (Q)$中公理化与$\mathrm{ZFC}$无关，其中$Q$是量词“存在不可数多”.
英文摘要：
A Kaufmann model is an $ω_1$-like, recursively saturated, rather classless model of $\mathrm{PA}$ or $\mathrm{ZF}$. Such models were constructed by Kaufmann under the combinatorial principle $\diamondsuit_{ω_1}$ and Shelah showed they exist in $\mathrm{ZFC}$ by an absoluteness argument. Kaufmann models are an important witness to the incompactness of $ω_1$ similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly naïve question of whether such a model can be "killed" by forcing without collapsing $ω_1$. We show that the answer to this question is independent of $\mathrm{ZFC}$ and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of $\mathrm{ZFC}$ whether or not Kaufmann models can be axiomatized in the logic $L_{ω_1, ω} (Q)$ where $Q$ is the quantifier "there exists uncountably many".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
