{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:31:40Z","timestamp":1775525500147,"version":"3.50.1"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10693,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider extensions of Peano arithmetic suitable for doing some of nonstandard analysis, in which there is a predicate <jats:italic>N<\/jats:italic>(<jats:italic>x<\/jats:italic>) for an elementary initial segment, along with axiom schemes approximating <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-saturation. We prove that such systems have the same proof-theoretic strength as their natural analogues in second order arithmetic. We close by presenting an even stronger extension of Peano arithmetic, which is equivalent to ZF for arithmetic statements.<\/jats:p>","DOI":"10.2307\/2274260","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:10:48Z","timestamp":1146953448000},"page":"1039-1058","source":"Crossref","is-referenced-by-count":13,"title":["The strength of nonstandard methods in arithmetic"],"prefix":"10.1017","volume":"49","author":[{"given":"C. Ward","family":"Henson","sequence":"first","affiliation":[]},{"given":"Matt","family":"Kaufmann","sequence":"additional","affiliation":[]},{"given":"H. Jerome","family":"Keisler","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200042353_ref009","first-page":"93","volume-title":"Applications of model theory to algebra, analysis, and probability","author":"Kreisel","year":"1969"},{"key":"S0022481200042353_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71123-6"},{"key":"S0022481200042353_ref005","unstructured":"Henson C. W. and Keisler H. 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