论文标题
集合理论的内部模型中的实数
The real numbers in inner models of set theory
论文作者
论文摘要
我们在集合理论的内部模型中研究真实的结构规律和不规则性。从$ L $（Gödel的可构造宇宙）开始，我们对真实的研究是两个方面的研究。一方面，我们研究了它们的生成过程与$ L $及其水平的属性如何相关，主要是指“可构造宇宙中的差距”（Marek and Srebrny，1974）。我们为该论文的结果提供了详细的证据，将它们推广到作者暗示的某些方向上，并通过介绍无限秩序差距的概念来介绍我们自己的概括，这是自然的，并产生了一些新的见解。另一方面，我们介绍并证明了一些众所周知的结果，这些结果构成了现实的病理集。我们将这项研究推广到$ l [\ #_ 1] $（在REAL的敏锐操作下关闭的最小内部模型）和$ L [\＃] $（在所有夏普下关闭的最小内部模型），为此，我们提供了一些介绍和基本事实，这些事实和基本事实在文献中不容易获得。我们还讨论了更大的内部模型的一些相关现代结果。
We study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, Gödel's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of $L$ and its levels, mainly referring to "Gaps in the constructible universe" (Marek and Srebrny, 1974). We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to $L[\#_1]$ (the smallest inner model closed under the sharp operation for reals) and $L[\#]$ (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models.
