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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,25]],"date-time":"2024-07-25T04:38:00Z","timestamp":1721882280680},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2021,2,16]],"date-time":"2021-02-16T00:00:00Z","timestamp":1613433600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2021,3]]},"abstract":"Abstract<\/jats:title>We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal

$M_1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, there is a model of

$\\mathsf {ZFC}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>in which

$\\mbox {NS}_{\\omega _{1}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is

$\\aleph _2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-saturated and

${\\Delta }_{1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-definable with

$\\omega _1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with

$\\mbox {NS}_{\\omega _{1}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>saturated and

${\\Delta }_{1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-definable with a ladder system

$\\vec {C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and a full Suslin treeT<\/jats:italic>as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article .<\/jats:p>","DOI":"10.1017\/jsl.2021.23","type":"journal-article","created":{"date-parts":[[2021,2,16]],"date-time":"2021-02-16T04:25:34Z","timestamp":1613449534000},"page":"25-59","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["NS SATURATED AND -DEFINABLE"],"prefix":"10.1017","volume":"86","author":[{"given":"STEFAN","family":"HOFFELNER","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2021,2,16]]},"reference":[{"key":"S0022481221000232_r9","doi-asserted-by":"publisher","DOI":"10.4064\/fm229-3-2"},{"key":"S0022481221000232_r1","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50024-0"},{"key":"S0022481221000232_r4","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2009.11.003"},{"key":"S0022481221000232_r23","doi-asserted-by":"crossref","DOI":"10.1017\/9781316717233","volume-title":"Proper and Improper Forcing","author":"Shelah","year":"2017"},{"key":"S0022481221000232_r18","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1993-030-6"},{"key":"S0022481221000232_r26","first-page":"1595","volume-title":"Handbook of Set Theory","author":"Shelah","year":"2010"},{"key":"S0022481221000232_r25","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/0168-0072(94)00021-T","article-title":"Projectively well-ordered inner models","volume":"74","author":"Shelah","year":"1995","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481221000232_r12","volume-title":"Set Theory","author":"Jech","year":"2003"},{"key":"S0022481221000232_r27","volume-title":"The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal","author":"Woodin","year":"2001"},{"key":"S0022481221000232_r10","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90004-3"},{"key":"S0022481221000232_r19","first-page":"1431","article-title":"On iterating semiproper preorders","volume":"67","author":"Miyamoto","year":"2002","journal-title":"this Journal"},{"key":"S0022481221000232_r22","unstructured":"[19] Schindler, R. , On $\\mathrm{NS}{\\omega}_1$ being saturated. Online notes. Available at http:\/\/www.math.uni-muenster.de\/u\/rds\/sat_ideal_better_version.pdf (accessed 20 June, 2016)."},{"key":"S0022481221000232_r13","first-page":"84","volume-title":"Mathematical Logic and Foundations of Set Theory","author":"Jensen","year":"1970"},{"key":"S0022481221000232_r7","volume-title":"Generalized Descriptive Set Theory","author":"Friedman","year":"2014"},{"key":"S0022481221000232_r2","first-page":"1126","article-title":"BPFA and projective wellorderings of the reals","volume":"76","author":"Caicedo","year":"2011","journal-title":"this Journal"},{"key":"S0022481221000232_r17","first-page":"1106","article-title":"${\\varSigma}_1(\\kappa)$ -definable subsets of $\\left({\\kappa}^{+}\\right)$","volume":"82","author":"L\u00fccke","year":"2017","journal-title":"this Journal"},{"key":"S0022481221000232_r11","first-page":"1677","article-title":"The canary tree revisited","volume":"66","author":"Hyttinen","year":"2001","journal-title":"this Journal"},{"key":"S0022481221000232_r8","unstructured":"[6] Friedman, S. D. and Wu, L. , Large cardinals and the ${\\varDelta}_1$ -definability of the nonstationary ideal. Preprint."},{"key":"S0022481221000232_r6","unstructured":"[8] Hoffelner, S. , Projective well-orders and the nonstationary ideal, Ph.D. thesis, University of Vienna, 2016."},{"key":"S0022481221000232_r15","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-004-0257-8"},{"key":"S0022481221000232_r3","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.2006.v13.n3.a5"},{"key":"S0022481221000232_r20","doi-asserted-by":"publisher","DOI":"10.4064\/fm-142-3-257-261"},{"key":"S0022481221000232_r21","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061305000407"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481221000232","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T15:36:00Z","timestamp":1671291360000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481221000232\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,16]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,3]]}},"alternative-id":["S0022481221000232"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2021.23","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,16]]},"assertion":[{"value":"\u00a9 The Association for Symbolic Logic 2021","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}