{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:39:08Z","timestamp":1777516748503,"version":"3.51.4"},"reference-count":25,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2016,7,13]],"date-time":"2016-07-13T00:00:00Z","timestamp":1468368000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2017,8,7]]},"abstract":"<jats:p>The separation between two theorems in reverse mathematics is usually done by constructing a Turing ideal satisfying a theorem P and avoiding the solutions to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a forcing technique for iterating a computable non-reducibility in order to separate theorems over omega-models. In this paper, we present a modularized version of their framework in terms of preservation of hyperimmunity and show that it is powerful enough to obtain the same separations results as Wang did with his notion of preservation of definitions. More than the actual separations, we provide a systematic method to design a computability-theoretic property which enables one to distinguish two statements, based on an analysis of their combinatorics.<\/jats:p>","DOI":"10.3233\/com-160062","type":"journal-article","created":{"date-parts":[[2016,7,15]],"date-time":"2016-07-15T12:23:21Z","timestamp":1468585401000},"page":"209-221","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":11,"title":["Iterative forcing and hyperimmunity in reverse mathematics"],"prefix":"10.1177","volume":"6","author":[{"given":"Ludovic","family":"Patey","sequence":"first","affiliation":[{"name":"Laboratoire PPS, Universit\u00e9 Paris Diderot, Paris, France. ."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2016,7,13]]},"reference":[{"key":"ref001","unstructured":"A.\u00a0Bovykin and A.\u00a0Weiermann, The strength of infinitary Ramseyan principles can be accessed by their densities,\n                      Annals of Pure and Applied Logic\n                      (2005), to appear, p.\u00a04."},{"key":"ref002","first-page":"104","volume":"21","author":"Cholak P.A.","year":"2001","journal-title":"Reverse Mathematics"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.2307\/2694910"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1254748693"},{"key":"ref005","unstructured":"S.\u00a0Flood and H.\u00a0Towsner, Separating principles below WKL\n                      0\n                      (2014), submitted, available at: http:\/\/arxiv.org\/abs\/1410.4068."},{"key":"ref006","unstructured":"H.M.\u00a0Friedman, Some systems of second order arithmetic and their use, in: Proceedings of the International Congress of Mathematicians, Vancouver, Vol.\u00a01, Canadian Mathematical Society, Montreal, Quebec, 1974, pp.\u00a0235\u2013242."},{"key":"ref007","unstructured":"H.M.\u00a0Friedman, Fom:53:free sets and reverse math and fom:54:recursion theory and dynamics, http:\/\/www.math.psu.edu\/simpson\/fom\/, available at: https:\/\/www.cs.nyu.edu\/pipermail\/fom\/."},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1142\/9208"},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061316500021"},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1174668391"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04847-8"},{"key":"ref012","first-page":"33","volume":"173","author":"Jockusch C.G.","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19930390153"},{"key":"ref014","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061313500074"},{"key":"ref015","unstructured":"L.\u00a0Patey, Combinatorial weaknesses of Ramseyan principles (2015), in preparation, available at: http:\/\/ludovicpatey.com\/media\/research\/combinatorial-weaknesses-draft.pdf."},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2015.07.003"},{"key":"ref017","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-015-0448-5"},{"key":"ref018","unstructured":"L.\u00a0Patey, Somewhere over the rainbow Ramsey theorem for pairs (2015), submitted, available at: http:\/\/arxiv.org\/abs\/1501.07424."},{"key":"ref019","unstructured":"L.\u00a0Patey, The strength of the tree theorem for pairs in reverse mathematics,\n                      Journal of Symbolic Logic\n                      (2015), to appear, available at: http:\/\/arxiv.org\/abs\/1505.01057."},{"key":"ref020","unstructured":"L.\u00a0Patey, Controlling iterated jumps of solutions to combinatorial problems,\n                      Computability\n                      (2016), to appear, available at: http:\/\/arxiv.org\/abs\/1509.05340."},{"key":"ref021","unstructured":"L.\u00a0Patey, The weakness of being cohesive, thin or free in reverse mathematics,\n                      Israel Journal of Mathematics\n                      (2016), to appear, available at: http:\/\/arxiv.org\/abs\/1502.03709."},{"key":"ref022","unstructured":"L.\u00a0Patey, A note on \u201cSeparating principles below Ramsey\u2019s theorem for pairs\u201d (2013), unpublished, available at: http:\/\/ludovicpatey.com\/media\/research\/note-em-sts.pdf."},{"key":"ref023","unstructured":"B.\u00a0Rice, Thin set for pairs implies DNR,\n                      Notre Dame J. Formal Logic\n                      , to appear."},{"key":"ref024","unstructured":"W.\u00a0Wang, The definability strength of combinatorial principles (2014), to appear, available at: http:\/\/arxiv.org\/abs\/1408.1465."},{"key":"ref025","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2014.05.003"}],"container-title":["Computability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-160062","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/COM-160062","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-160062","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T15:59:53Z","timestamp":1777391993000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-160062"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7,13]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,8,7]]}},"alternative-id":["10.3233\/COM-160062"],"URL":"https:\/\/doi.org\/10.3233\/com-160062","relation":{},"ISSN":["2211-3568","2211-3576"],"issn-type":[{"value":"2211-3568","type":"print"},{"value":"2211-3576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,7,13]]}}}