{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:15:32Z","timestamp":1753884932089,"version":"3.41.2"},"reference-count":25,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC","doi-asserted-by":"crossref","award":["EP\/N032160\/1"],"award-info":[{"award-number":["EP\/N032160\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2022,8,1]]},"abstract":"<jats:p> We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing\u00a0\u2014 in this situation [Formula: see text]\u00a0\u2014 to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0219061322500064","type":"journal-article","created":{"date-parts":[[2022,6,22]],"date-time":"2022-06-22T09:27:13Z","timestamp":1655890033000},"source":"Crossref","is-referenced-by-count":1,"title":["Incompatible bounded category forcing axioms"],"prefix":"10.1142","volume":"22","author":[{"given":"David","family":"Asper\u00f3","sequence":"first","affiliation":[{"name":"School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK"}]},{"given":"Matteo","family":"Viale","sequence":"additional","affiliation":[{"name":"Department of Mathematics \u201cGiuseppe Peano\u201d, University of Torino, via Carlo Alberto 10, 10125, Torino, Italy"}]}],"member":"219","published-online":{"date-parts":[[2022,6,22]]},"reference":[{"key":"S0219061322500064BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-62935-3_1"},{"key":"S0219061322500064BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2013.06.006"},{"key":"S0219061322500064BIB003","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2021.193.3.3"},{"key":"S0219061322500064BIB004","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1190150154"},{"key":"S0219061322500064BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061317500052"},{"key":"S0219061322500064BIB008","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2006.v13.n3.a5"},{"key":"S0219061322500064BIB009","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1058448434"},{"key":"S0219061322500064BIB010","doi-asserted-by":"publisher","DOI":"10.2307\/1971415"},{"key":"S0219061322500064BIB011","first-page":"307","volume":"6","author":"Goldstern M.","year":"1992","journal-title":"Israel Math. 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