{"title":"Bounds on Scott ranks of some polish metric spaces","authors":"William Chan","doi":"10.1142/s021906132150001x","DOIUrl":null,"url":null,"abstract":"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].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"108 1","pages":"2150001:1-2150001:23"},"PeriodicalIF":0.9000,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s021906132150001x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 2
Abstract
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].
如果[公式:见文]是一个适当的波兰度量空间，而[公式:见文]是[公式:见文]的任何可数稠密子度量空间，那么[公式:见文]在度量空间的自然一阶语言中的Scott秩是可数的，实际上至多[公式:见文]，其中[公式:见文]是[公式:见文]的Church-Kleene序数(解释为[公式:见文]的一个子集)，它是没有表示的最小序数[公式:见文]:可由[公式:见文本]计算的。如果[Formula: see text]是一个严格的波兰度量空间，而[Formula: see text]是任何可数的密集子度量空间，那么[Formula: see text]的Scott秩是可数的，实际上小于[Formula: see text]。
期刊介绍：
The Journal of Mathematical Logic (JML) provides an important forum for the communication of original contributions in all areas of mathematical logic and its applications. It aims at publishing papers at the highest level of mathematical creativity and sophistication. JML intends to represent the most important and innovative developments in the subject.
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