论文标题
用尿液元素在集合理论中的公理化和强迫
Axiomatization and Forcing in Set Theory with Urelements
论文作者
论文摘要
在本文的第一部分中，我们考虑了尿门集合理论中的几个自然公理，包括收集原理，反思原理，依赖选择方案及其概括以及其他专门针对尿液元素的公理。我们证明，这些公理在$ \ ZFCUR $（ZFC（ZFC）构成了静脉结构，并在直接含义上形成了带有尿液的尿液元素）。论文研究的第二部分迫使$ \ zfur $的可计数转移模型。我们提出了$¶$名称的新定义，以通过现有方法解决问题。然后，我们证明了有关公理保存的尿液元素强迫的基本定理。此外，我们表明强迫可以在先前建立的层次结构内销毁和恢复某些公理。最后，我们证明了接地模型包含适当的尿液元素时，地面模型可确定性如何失败。
In the first part of this paper, we consider several natural axioms in urelement set theory, including the Collection Principle, the Reflection Principle, the Dependent Choice scheme and its generalizations, as well as other axioms specifically concerning urelements. We prove that these axioms form a hierarchy over $\ZFCUR$ (ZFC with urelements formulated with Replacement) in terms of direct implication. The second part of the paper studies forcing over countable transitive models of $\ZFUR$. We propose a new definition of $¶$-names to address an issue with the existing approach. We then prove the fundamental theorem of forcing with urelements regarding axiom preservation. Moreover, we show that forcing can destroy and recover certain axioms within the previously established hierarchy. Finally, we demonstrate how ground model definability may fail when the ground model contains a proper class of urelements.
