{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T15:33:21Z","timestamp":1767713601738},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>This paper compares the strength of two sorts of sentences of PA (classical first-order arithmetic with induction): reflection principles and sentences that may be called iterated consistency assertions.<\/jats:p><jats:p>Let Bew(<jats:italic>x<\/jats:italic>) be the standard provability predicate for PA, and for any sentence <jats:italic>S<\/jats:italic> of PA, let \u2308<jats:italic>S<\/jats:italic>\u2309 be the numeral for the G\u00f6del number of <jats:italic>S. The reflection principle for S<\/jats:italic> is the sentence Bew(\u2308<jats:italic>S<\/jats:italic>\u2309) \u2192 <jats:italic>S<\/jats:italic>, and a <jats:italic>reflection principle<\/jats:italic> is simply the reflection principle for some sentence. Nothing false (in the standard model for PA) is provable in PA, and therefore every reflection principle is true. L\u00f6b's theorem asserts that <jats:italic>S<\/jats:italic> is provable (in PA) if the reflection principle for <jats:italic>S<\/jats:italic> is provable.<\/jats:p><jats:p>We shall suppose that the 0-ary propositional connectives \u22a4 and \u22a5 are taken as primitives in the formulation of PA. We define the <jats:italic>iterated consistency assertions<\/jats:italic> Con<jats:sup><jats:italic>m<\/jats:italic><\/jats:sup> by: Con<jats:sup>0<\/jats:sup> = \u22a4; Con<jats:sup><jats:sub>m<\/jats:sub>\u22121<\/jats:sup> = \u2212 Bew(\u2308 \u2212 Con<jats:sup><jats:italic>m<\/jats:italic><\/jats:sup>\u2309). Con<jats:sup>1<\/jats:sup> may be taken to be the sentence of PA that expresses the consistency of PA; Con<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup>, the sentence that expresses the consistency of PA \u22c3 {Con<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>}.<\/jats:p><jats:p>Our starting point is the observation that Con<jats:sup>1<\/jats:sup> is equivalent (in PA) to the reflection principle for \u22a5. (The second incompleteness theorem thus follows in a well-known way from L\u00f6b's theorem: if PA is consistent, then \u22a5 is not provable, the reflection principle for \u22a5 is not provable, and the consistency of PA is not provable either.)<\/jats:p>","DOI":"10.2307\/2273701","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:49:34Z","timestamp":1146952174000},"page":"33-35","source":"Crossref","is-referenced-by-count":9,"title":["Reflection principles and iterated consistency assertions"],"prefix":"10.1017","volume":"44","author":[{"given":"George","family":"Boolos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048696_ref002","volume-title":"Filosofiska F\u00f6reningen och Filosofiska Institutionem vid Uppsala Universitet, Uppsala","author":"Segerberg","year":"1971"},{"key":"S0022481200048696_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757006"},{"key":"S0022481200048696_ref001","first-page":"115","volume":"20","author":"L\u00f6b","year":"1955","journal-title":"Solution of a problem of Leon Henkin"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048696","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T21:56:51Z","timestamp":1558907811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048696\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":3,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048696"],"URL":"https:\/\/doi.org\/10.2307\/2273701","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}