Hjorth (Variations on Scott, 1998; The fine structure and Borel complexity of orbits, 2010) introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth’s question positively. As a corollary we prove the following conjecture of Hjorth—for every limit ordinal \(\alpha \), the set of elements whose orbit is of complexity less than \(\alpha \) is a Borel set.

中文翻译：

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Hjorth 秩的有界原理

Hjorth（Scott 的变体，1998 年；轨道的精细结构和博雷尔复杂性，2010 年）介绍了对一般波兰群行为的 Scott 分析，并询问他的秩概念是否满足类似于 Scott 秩的有界原则——即，如果轨道等价关系为 Borel，则 Hjorth 秩是有界的。我们正面回答 Hjorth 的问题。作为推论，我们证明了以下 Hjorth 猜想——对于每个极限序数\(\alpha \)，其轨道复杂度小于\(\alpha \)的元素集是一个 Borel 集。