论文标题
每个可数算术或集合理论的可数模型都有一个可定义的末端扩展
Every countable model of arithmetic or set theory has a pointwise-definable end extension
论文作者
论文摘要
根据数学茶的论点，必须有我们无法描述或定义的实数，因为有很多实数，但只有很多定义。然而，在没有参数的情况下，每个人都可以定义的集合理论模型的存在挑战了这一结论。在本文中，我介绍了一种灵活的新方法，用于构建算术和集合理论的可点上可定义的模型，并表明每个可计数的Zermelo-fraenkel ZF ZF Set理论和Peano Arithmetic PA的可数模型都具有可点击的最终扩展。在算术案例中，我使用通用算法及其$σ_n$概括来构建一个逐步的基本塔，从而使每个阶段$ n $都可以定义任何所需的个人$ a_n $，同时将这些定义保留到限制模型，因此可以将这些定义安排为可以定义的。类似的方法在集合理论中起作用，而且可以在扩展中实现$ v = l $，或者实际上在原始模型的内部模型中保存的任何其他合适的理论，从而实现了复活现象。例如，每个具有可测量基数的内部模型的ZF的可数模型都具有$ \ text {zfc}+v = l [μ] $的可点式定义模型的末端扩展。
According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its $Σ_n$ generalizations to build a progressively elementary tower making any desired individual $a_n$ definable at each stage $n$, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve $V=L$ in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of $\text{ZFC}+V=L[μ]$.
