{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,1,29]],"date-time":"2025-01-29T05:37:20Z","timestamp":1738129040444,"version":"3.33.0"},"reference-count":18,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T00:00:00Z","timestamp":1698192000000},"content-version":"vor","delay-in-days":1,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,2,27]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The notions of permutable and weak-permutable convergence of a series $\\sum _{n=1}^{\\infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann\u2019s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara\u2019s principle BD- $\\mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $\\mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $\\mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.<\/jats:p>","DOI":"10.1093\/jigpal\/jzad024","type":"journal-article","created":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T23:17:57Z","timestamp":1698275877000},"page":"49-61","source":"Crossref","is-referenced-by-count":0,"title":["Constructive aspects of Riemann\u2019s permutation theorem for series"],"prefix":"10.1093","volume":"33","author":[{"given":"J","family":"Berger","sequence":"first","affiliation":[{"name":"Department of Mathematics , University of Munich, Theresienstra\u00dfe 39, 80333 Munich,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Douglas","family":"Bridges","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , University of Canterbury, Christchurch 8041,","place":["New Zealand"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hannes","family":"Diener","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , University of Canterbury, Christchurch 8041,","place":["New Zealand"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Helmet","family":"Schwichtenberg","sequence":"additional","affiliation":[{"name":"Department of Mathematics , University of Munich, Theresienstra\u00dfe 39, 80333 Munich,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2023,10,24]]},"reference":[{"volume-title":"Constructive Set Theory","author":"Aczel","key":"2025012810511244700_ref1"},{"key":"2025012810511244700_ref2","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.tcs.2022.06.019","article-title":"Morse set theory as a foundation for constructive mathematics","volume":"928","author":"Bridges","year":"2022","journal-title":"Theor. 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A Chapter of Intuitionistic Mathematics","year":"1977","author":"Troelstra","key":"2025012810511244700_ref18"}],"container-title":["Logic Journal of the IGPL"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/33\/1\/49\/52459179\/jzad024.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jigpal\/article-pdf\/33\/1\/49\/52459179\/jzad024.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T10:51:24Z","timestamp":1738061484000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jigpal\/article\/33\/1\/49\/7329113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,24]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2,27]]}},"URL":"https:\/\/doi.org\/10.1093\/jigpal\/jzad024","relation":{},"ISSN":["1367-0751","1368-9894"],"issn-type":[{"type":"print","value":"1367-0751"},{"type":"electronic","value":"1368-9894"}],"subject":[],"published-other":{"date-parts":[[2025,2]]},"published":{"date-parts":[[2023,10,24]]}}}