{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T08:18:09Z","timestamp":1769156289382,"version":"3.49.0"},"reference-count":34,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2018,12,27]],"date-time":"2018-12-27T00:00:00Z","timestamp":1545868800000},"content-version":"vor","delay-in-days":26,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"name":"PSC CUNY","award":["60630-00 48"],"award-info":[{"award-number":["60630-00 48"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2018,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is shown that the boldface maximality principle for subcomplete forcing, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0001\"\/>, together with the assumption that the universe has only set many grounds, implies the existence of a well\u2010ordering of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0002\"\/> definable without parameters. The same conclusion follows from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0003\"\/>, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0004\"\/> does not exist, for some <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0005.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0005\"\/>, implies the existence of a well\u2010ordering of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0006.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0006\"\/> which is \u0394<jats:sub>1<\/jats:sub>\u2010definable without parameters, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0007.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0007\"\/>\u2010definable using a subset of \u03c9<jats:sub>1<\/jats:sub> as a parameter. This well\u2010order is in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0008.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0008\"\/>. Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201800008-math-0009.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201800008:malq201800008-math-0009\"\/> mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.<\/jats:p>","DOI":"10.1002\/malq.201800008","type":"journal-article","created":{"date-parts":[[2018,12,27]],"date-time":"2018-12-27T11:06:40Z","timestamp":1545908800000},"page":"487-504","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Subcomplete forcing principles and definable well\u2010orders"],"prefix":"10.1002","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4627-3154","authenticated-orcid":false,"given":"Gunter","family":"Fuchs","sequence":"first","affiliation":[{"name":"Department of Mathematics College of Staten Island, City University of New York 2800 Victory Boulevard Staten Island NY 10314 United States of America"},{"name":"The Graduate Center City University of New York 365 5th Avenue New York NY 10016 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2018,12,27]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050154"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-011-0261-8"},{"key":"e_1_2_7_4_1","volume-title":"Topics in Set Theory. 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