集合论的模伴

摘要：

这是一个介绍性的文件，一系列的结果，连接通用的绝对性的结果，二阶和三阶数论的模型理论的概念模型的伴侣。具体来说，我们在这里开发了一个一般框架，将Woodin的二阶数论的一般绝对性结果和普遍Baire集的理论联系起来，以建立伴侣关系，并表明，（详细说明所需注意事项）一个$Π_2$ -用适当的语言形式化的二阶数论的性质是强制的，当且仅当它与$T\supseteq\mathsf{ZFC}+$的通用片段一致，如果并且只有在$T$的模型伴侣中实现。特别是，我们证明了$H_{ω_1}$的一阶理论是集合论域的一阶理论的模型伴侣，假设存在多个Woodin基数类，并在签名中使用针对$Δ_0$ -属性和所有普遍贝尔实数集的谓词。我们将这些结果也推广到$H_{\aleph_2}$的理论在本文的后续行动。

英文摘要：

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $Π_2$-property formalized in an appropriate language for second 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 $T$. In particular we show that the first order theory of $H_{ω_1}$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $Δ_0$-properties and for all universally Baire sets of reals. We will extend these results also to the theory of $H_{\aleph_2}$ in a follow up of this paper.

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。