{"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":"40 1","pages":"168 - 186"},"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}
引用次数: 1
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,
$\Sigma ^1_1$
-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.