论文标题
设置理论$ i $的驯服
Tameness for set theory $I$
论文作者
论文摘要
该论文是两者中的第一个，旨在表明（假设大型红衣主教集理论理论是一种可进行的（并且我们敢于说驯服）一阶理论，当时在一阶签名中正式化，具有自然谓词符号，用于自然谓词符号，用于第二阶和三阶算术的基本可定义概念，并吸引了模型的模型和模型完整性和模型完整性和模型概念的概念。 具体而言，我们开发了一个一般框架，将一般的绝对性结果连接到建模陪伴，并表明（详细介绍了所需的护理）$π_2$ property，以适当的语言进行第二或三阶数字理论形式化，可以从某些$ t \ supseteq \ mathsf {zfc}+$ giald Cartinals If unction unction unign $ night In n IN $ primn $ nignign $ trign $ trign $ trign $ trign $ trign $ priff n n In tright $ trightim property。 $ t $的同伴。 任何在这两个主题的本科课程层面上都有公平熟悉的人和一阶逻辑的任何人都可以访问该论文。但是，这可能会出现怪异（鉴于我们旨在证明的结果）不需要强迫或大型红衣主教才能获得其主要结果的证明（如果一个人接受黑盒，则相关的通用绝对性结果）。另一方面，熟悉模型完整性和模型同伴的概念至关重要。所有必要的模型理论背景将详细介绍。 目前的工作扩大并系统化了Venturi获得的先前结果。
The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $Π_2$-property formalized in an appropriate language for second or third order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large cardinals if and only if it is consistent with the universal fragment of $T$ if and only if it is realized in the model companion of $T$. The paper is accessible to any person who has a fair acquaintance with set theory and first order logic at the level of an under-graduate course in both topics; however bizarre this may appear (given the results we aim to prove) no knowledge of forcing or large cardinals is required to get the proofs of its main results (if one accepts as black-boxes the relevant generic absoluteness results). On the other hand familiarity with the notions of model completeness and model companionship is essential. All the necessary model-theoretic background will be given in full detail. The present work expands and systematize previous results obtained with Venturi.
