{"title":"On the relationships between some meta-mathematical properties of arithmetical theories","authors":"Yong Cheng","doi":"10.1093/jigpal/jzad015","DOIUrl":null,"url":null,"abstract":"\n In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\\textbf{0}^{\\prime }$ (theories with Turing degree $\\textbf{0}^{\\prime }$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ in the above list, we examine whether $P$ implies $Q$.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Logic Journal of the IGPL","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/jigpal/jzad015","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 1
Abstract
In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\textbf{0}^{\prime }$ (theories with Turing degree $\textbf{0}^{\prime }$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ in the above list, we examine whether $P$ implies $Q$.
期刊介绍:
Logic Journal of the IGPL publishes papers in all areas of pure and applied logic, including pure logical systems, proof theory, model theory, recursion theory, type theory, nonclassical logics, nonmonotonic logic, numerical and uncertainty reasoning, logic and AI, foundations of logic programming, logic and computation, logic and language, and logic engineering.
Logic Journal of the IGPL is published under licence from Professor Dov Gabbay as owner of the journal.