论文标题
Kaufmann模型的可耐受性和公理性
Destructibility and Axiomatizability of Kaufmann Models
论文作者
论文摘要
Kaufmann模型是$ω_1$的，类似于递归饱和的，而不是$ \ mathrm {pa} $或$ \ mathrm {zf} $的无类模型。这样的模型是由考夫曼根据组合原理构建的$ \ diamondsuit_ {ω_1} $，而谢拉（Shelah）则显示它们存在于$ \ mathrm {zfc} $中，这是绝对的参数。 Kaufmann型号是与Aronszajn树相似的$ω_1$不兼容性的重要见证。在本文中，我们研究了一些与此相关的设定理论问题，该问题是出于看似幼稚的问题，即是否可以通过强制崩溃$ω_1$来“杀死”这种模型。我们表明，这个问题的答案独立于$ \ mathrm {zfc} $，并且与有关Aronszajn树的类似问题密切相关。作为这些方法的应用，我们还表明，它与$ \ mathrm {zfc} $无关，无论是否可以在逻辑$ l_ {ω_1，ω_}（q）$中对kaufmann型号进行公理，其中$ q $是量子，$ q $是量化的，这是量不高的。
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".
