{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T05:40:17Z","timestamp":1695879617611},"reference-count":2,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":5430,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let \u0393<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(\u03c6) be a formula of L<jats:sub>PA<\/jats:sub> (PA = Peano Arithmetic) meaning \u201cthere is a proof of \u03c6 from PA\u2010axioms, in which \u03c9\u2010rule is iterated no more than <jats:italic>n<\/jats:italic> times\u201d. We examine relations over pairs of natural numbers of the kind.<\/jats:p><jats:p>(<jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>) \u2266<jats:sub><jats:italic>H<\/jats:italic><\/jats:sub> (<jats:italic>n', k<\/jats:italic>') iff PA + RFN<jats:sub><jats:italic>n<\/jats:italic>'<\/jats:sub> (<jats:italic>H<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>'<\/jats:sub>) \u22a9 RFN<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> (<jats:italic>H<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>).<\/jats:p><jats:p>Where <jats:italic>H<\/jats:italic> denotes one of the hierarchies \u2211 or \u03a0 and RFN<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>C<\/jats:italic>) is the scheme of the reflection principle for \u0393<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> restricted to formulas from the class <jats:italic>C<\/jats:italic>(\u0393<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(\u03c6) implies \u201c\u03c6 is true\u201d, for every \u03c6 \u2208 <jats:italic>C<\/jats:italic>). Our main result is that.<\/jats:p><jats:p>(<jats:italic>n<\/jats:italic>, <jats:italic>k<\/jats:italic>) \u2266<jats:italic>H<\/jats:italic> (<jats:italic>n', k<\/jats:italic>') if <jats:italic>n<\/jats:italic> \u2266 <jats:italic>n<\/jats:italic>' and <jats:italic>k<\/jats:italic> \u2266 max (<jats:italic>k', 2n<\/jats:italic>' + 1).<\/jats:p>","DOI":"10.1002\/malq.19920380116","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T05:44:57Z","timestamp":1180417497000},"page":"203-208","source":"Crossref","is-referenced-by-count":1,"title":["ON THE ITERATED \u03c9\u2010RULE"],"prefix":"10.1002","volume":"38","author":[{"given":"Grzegorz","family":"Michalski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"531","article-title":"Bounded induction and satisfaction classes","volume":"32","author":"Kotlarski H.","year":"1986","journal-title":"This Zeitschrift"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71123-6"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19920380116","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19920380116","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T20:35:57Z","timestamp":1695846957000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19920380116"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,1]]},"references-count":2,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,1]]}},"alternative-id":["10.1002\/malq.19920380116"],"URL":"https:\/\/doi.org\/10.1002\/malq.19920380116","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,1]]}}}