发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带本元的一般形式迭代集合概念，构建阶段与层级两种理论，发现其不等价，引入有向性等原则可恢复等价性并产生拟范畴性，揭示更强集合概念。
AI 中文摘要
我们以最一般的形式研究集合的迭代概念，允许存在本元（urelements），且不假设这些本元构成一个集合。我们将该概念表述为两种形式：阶段理论（stage theory）和层级理论（level theory），并建立了带本元的一般层级理论。与纯集合对应的理论不同，所得的阶段理论和层级理论在集合论上并不等价；此外，带本元的二阶层级理论并非弱拟范畴的。随后，我们基于有向性、无界性和反思性，考虑了进一步支配阶段和层级的原则。部分此类原则恢复了对应理论间的集合论等价性，而其层级理论版本则产生了拟范畴性的形式。这些原则构成严格蕴含层级，从而揭示了超越基本迭代概念的不同更强集合概念。
英文摘要
We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.
