{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,17]],"date-time":"2023-09-17T19:42:09Z","timestamp":1694979729158},"reference-count":12,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2018,7,3]],"date-time":"2018-07-03T00:00:00Z","timestamp":1530576000000},"content-version":"vor","delay-in-days":2,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2018,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that the Boolean Prime Ideal Theorem (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0001\" \/>) does not imply the Nielsen\u2010Schreier Theorem (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0002\" \/>) in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0003\" \/>, thus strengthening the result of Kleppmann from \u201cNielsen\u2010Schreier and the Axiom of Choice\u201d that the (strictly weaker than <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0004\" \/>) Ordering Principle (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0005\" \/>) does not imply <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0006\" \/> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0007\" \/>. We also show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0008\" \/> is false in Mostowski's Linearly Ordered Model of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201700022-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201700022:malq201700022-math-0009\" \/>. The above two results also settle the corresponding open problems from Howard and Rubin's \u201cConsequences of the Axiom of Choice\u201d.<\/jats:p>","DOI":"10.1002\/malq.201700022","type":"journal-article","created":{"date-parts":[[2018,7,4]],"date-time":"2018-07-04T06:06:20Z","timestamp":1530684380000},"page":"173-177","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A note on the deductive strength of the Nielsen\u2010Schreier theorem"],"prefix":"10.1002","volume":"64","author":[{"given":"Eleftherios","family":"Tachtsis","sequence":"first","affiliation":[{"name":"Department of Mathematics University of the Aegean Karlovassi 83200 Samos Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,7,3]]},"reference":[{"key":"e_1_2_6_2_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-127-1-57-66"},{"key":"e_1_2_6_3_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-55-1-57-66"},{"key":"e_1_2_6_4_1","doi-asserted-by":"publisher","DOI":"10.2307\/2274234"},{"key":"e_1_2_6_5_1","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_6_6_1","volume-title":"Studies in Logic and the Foundations of Mathematics","author":"Jech T. 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