{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T12:37:08Z","timestamp":1765370228319,"version":"3.44.0"},"reference-count":19,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2023,7,18]],"date-time":"2023-07-18T00:00:00Z","timestamp":1689638400000},"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":[[2023,11,13]]},"abstract":"<jats:p> In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schr\u00f6der\u2013Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice. <\/jats:p>","DOI":"10.3233\/com-230453","type":"journal-article","created":{"date-parts":[[2023,7,18]],"date-time":"2023-07-18T11:33:12Z","timestamp":1689679992000},"page":"203-225","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Banach\u2019s theorem in higher-order reverse mathematics"],"prefix":"10.1177","volume":"12","author":[{"given":"Jeffry\u00a0L.","family":"Hirst","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Appalachian State University, Boone, NC, USA"}]},{"given":"Carl","family":"Mummert","sequence":"additional","affiliation":[{"name":"Department of Computer and Information Technology, Marshall University, Huntington, WV, USA"}]}],"member":"179","published-online":{"date-parts":[[2023,7,18]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-6-1-236-239"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1294170993"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-11367-3"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71126-1"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2270259"},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-86896-2"},{"key":"ref007","unstructured":"J.L.\u00a0Hirst, Combinatorics in subsystems of second order arithmetic, PhD thesis, The Pennsylvania State University, 1987. http:\/\/gateway.proquest.com\/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi\/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:8728018."},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/106\/1057822"},{"key":"ref009","unstructured":"J.\u00a0Hunter, Higher-order reverse topology, PhD thesis, The University of Wisconsin\u2013Madison, 2008. 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