If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] (construed as a subset of [Formula: see text]) which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].

中文翻译：

---

一些波兰度量空间的斯科特等级的界限

如果 [Formula: see text] 是一个适当的波兰度量空间，并且 [Formula: see text] 是 [Formula: see text] 的任何可数稠密子度量空间，那么 [Formula: see text] 的 Scott 秩在自然第一-度量空间的顺序语言是可数的，实际上最多 [Formula: see text]，其中 [Formula: see text] 是 [Formula: see text] 的 Church-Kleene 序数（解释为 [Formula: see text] 的子集]) 这是最小的序数，在 [Formula: see text] 上没有表示可从 [Formula: see text] 计算。如果 [Formula: see text] 是刚性波兰度量空间并且 [Formula: see text] 是任何可数稠密子度量空间，则 [Formula: see text] 的 Scott 秩是可数的并且实际上小于 [Formula: see text ]。