Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem

A game on the universe of sets

In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are winning, i.e. have a winning strategy for a player, form a natural hierarchy with levels indexed by ordinals. We show that the class of hereditarily winning sets is an inner model containing all well-founded sets, and that all four possible relationships between the universe, the class of hereditarily winning sets, and the class of well-founded sets are consisten…

终端余代数与同伦类型理论中的非良基础集

Terminal Coalgebras and Non-wellfounded Sets in Homotopy Type Theory

本文在同伦类型理论中构建了非良基础集模型，提出扩展AFA和SAFA的通用化，并通过M型构建了满足SAFA通用化的层级模型，同时刻画了M型的身份类型。

A Cook's Tour of the Finitary Non-Well-Founded Sets

We give multiple descriptions of a topological universe of finitary sets, which can be seen as a natural limit completion of the hereditarily finite sets. This universe is characterized as a metric completion of the hereditarily finite sets; as a Stone space arising as the solution of a functorial fixed-point equation involving the Vietoris construction; as the Stone dual of the free modal algebra; and as the subspace of maximal elements of a domain equation involving the Plotkin (or convex) powerdomain. These results illustrate the methods developed in the author's 'Domain theory in logical f…

Arboreal范畴：一种资源的公理化理论

Arboreal Categories: An Axiomatic Theory of Resources

本文提出arboreal categories的公理化框架，通过内在过程结构和arboreal covers捕获余代数范畴的本质属性，统一并恢复了有限模型论与描述复杂性中的基本概念。