{"title":"基于参数化ackermann-pÉter函数的Goodstein序列","authors":"T. Arai, S. Wainer, A. Weiermann","doi":"10.1017/bsl.2021.30","DOIUrl":null,"url":null,"abstract":"Abstract Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA, \n$\\Sigma ^1_1$\n -DC \n$_0$\n , ATR \n$_0$\n , up to ID \n$_1$\n . The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal form representations of positive integers.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"GOODSTEIN SEQUENCES BASED ON A PARAMETRIZED ACKERMANN–PÉTER FUNCTION\",\"authors\":\"T. Arai, S. Wainer, A. Weiermann\",\"doi\":\"10.1017/bsl.2021.30\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA, \\n$\\\\Sigma ^1_1$\\n -DC \\n$_0$\\n , ATR \\n$_0$\\n , up to ID \\n$_1$\\n . The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal form representations of positive integers.\",\"PeriodicalId\":22265,\"journal\":{\"name\":\"The Bulletin of Symbolic Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Bulletin of Symbolic Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/bsl.2021.30\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.30","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
GOODSTEIN SEQUENCES BASED ON A PARAMETRIZED ACKERMANN–PÉTER FUNCTION
Abstract Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA,
$\Sigma ^1_1$
-DC
$_0$
, ATR
$_0$
, up to ID
$_1$
. The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal form representations of positive integers.