In constructive set theory, an ordinal is a hereditarily transitive set. In homotopy type theory (HoTT), an ordinal is a type with a transitive, wellfounded, and extensional binary relation. We show that the two definitions are equivalent if we use (the HoTT refinement of) Aczel's interpretation of constructive set theory into type theory. Following this, we generalize the notion of a type-theoretic ordinal to capture all sets in Aczel's interpretation rather than only the ordinals. This leads to a natural class of ordered structures which contains the type-theoretic ordinals and realizes the higher inductive interpretation of set theory. All our results are formalized in Agda.

翻译：在建设性设定理论中,一个正弦是一个异端的中转集体。在同质类型理论(Hott)中,正弦是一个具有过渡性、有充分根据和扩展性二进制关系的类型。我们表明,如果我们使用(Hott对Aczel对建设性设定理论的解释进行)Aczel的典型理论解释为类型理论,那么这两个定义就等同了。在此之后,我们推广了一种类型理论或dinal的概念,以捕捉Aczel解释中的所有组合,而不仅仅是正弦。这导致了一种包含类型理论的自然定序结构,并实现了对设定理论的更高感化解释。我们的所有结果都在阿格达正式化。