{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T05:12:04Z","timestamp":1762060324523,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T00:00:00Z","timestamp":1658448000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research RFBR","doi-asserted-by":"publisher","award":["20-01-00670"],"award-info":[{"award-number":["20-01-00670"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing notions to define a model of set theory ZFC in which, for a given n&gt;2, there exists a good \u0394n1 well-ordering of the reals but there are no such well-orderings in the class \u0394n\u221211. Therefore the existence of a well-ordering of the reals at a certain level n&gt;2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n\u22121. This is a new result in such a generality (with n&gt;2 arbitrary), and it may lead to further progress in studies of the projective hierarchy.<\/jats:p>","DOI":"10.3390\/axioms11080354","type":"journal-article","created":{"date-parts":[[2022,7,22]],"date-time":"2022-07-22T12:53:45Z","timestamp":1658494425000},"page":"354","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Model in Which Well-Orderings of the Reals Appear at a Given Projective Level"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7415-9784","authenticated-orcid":false,"given":"Vladimir","family":"Kanovei","sequence":"first","affiliation":[{"name":"Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3739-9161","authenticated-orcid":false,"given":"Vassily","family":"Lyubetsky","sequence":"additional","affiliation":[{"name":"Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"261","DOI":"10.24033\/bsmf.761","article-title":"Cinq lettres sur la th\u00e9orie des ensembles","volume":"33","author":"Hadamard","year":"1905","journal-title":"Bull. Soc. Math. Fr."},{"key":"ref_2","first-page":"91","article-title":"Sur la classification de M. Baire","volume":"164","author":"Lusin","year":"1917","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_3","unstructured":"Sierpi\u0144ski, W. (1918). L\u2019axiome de M. Zermelo et son r\u00f4le dans la th\u00e9orie des ensembles et l\u2019analyse. Bull. Acad. Sci. Cracovie, 97\u2013152."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kechris, A.S. (1995). Classical Descriptive Set Theory, Springer.","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"ref_5","first-page":"XII","article-title":"Descriptive set theory","volume":"Volume 100","author":"Moschovakis","year":"1980","journal-title":"Studies in Logic and the Foundations of Mathematics"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"G\u00f6del, K. (1940). The Consistency of the Continuum Hypothesis, Princeton University Press. Annals of Mathematics Studies, no. 3.","DOI":"10.1515\/9781400881635"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"337","DOI":"10.4064\/fm-46-3-337-357","article-title":"Some consequences of the axiom of constructibility","volume":"46","author":"Addison","year":"1959","journal-title":"Fundam. Math."},{"key":"ref_8","unstructured":"Bar-Hillel, Y. (1970). Definability in Axiomatic Set Theory II. Studies in Logic and the Foundations of Mathematics, North-Holland."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1970696","article-title":"A model of set-theory in which every set of reals is Lebesgue measurable","volume":"92","author":"Solovay","year":"1970","journal-title":"Ann. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.2178\/jsl\/1318338842","article-title":"BPFA and projective well-orderings of the reals","volume":"76","author":"Caicedo","year":"2011","journal-title":"J. Symb. Log."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1016\/j.apal.2009.11.003","article-title":"Cardinal characteristics and projective wellorders","volume":"161","author":"Fischer","year":"2010","journal-title":"Ann. Pure Appl. Logic"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1016\/j.apal.2012.12.001","article-title":"Cardinal characteristics, projective wellorders and large continuum","volume":"164","author":"Fischer","year":"2013","journal-title":"Ann. Pure Appl. Log."},{"key":"ref_13","first-page":"1","article-title":"Measure, category and projective wellorders","volume":"6","author":"Fischer","year":"2014","journal-title":"J. Log. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1466","DOI":"10.1017\/jsl.2019.43","article-title":"A \u03a341 wellorder of the reals with NS\u03c91 saturated","volume":"84","author":"Friedman","year":"2019","journal-title":"J. Symb. Log."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/S0049-237X(08)71934-7","article-title":"Definable sets of minimal degree","volume":"Volume 59","year":"1970","journal-title":"Studies in Logic and the Foundations of Mathematics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.apal.2020.102929","article-title":"The full basis theorem does not imply analytic wellordering","volume":"172","author":"Kanovei","year":"2021","journal-title":"Ann. Pure Appl. Log."},{"key":"ref_17","first-page":"x","article-title":"Fine Structure and Class Forcing","volume":"Volume 3","author":"Friedman","year":"2000","journal-title":"De Gruyter Series in Logic and Its Applications"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/978-1-4020-5764-9_9","article-title":"Constructibility and class forcing","volume":"Volume 3","author":"Friedman","year":"2010","journal-title":"Handbook of Set Theory"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1070\/IM8937","article-title":"Models of set theory in which separation theorem fails","volume":"85","author":"Kanovei","year":"2021","journal-title":"Izv. Math."},{"key":"ref_20","unstructured":"Jech, T. (2003). Set Theory, Springer. [The Third Millennium Revised and Expanded ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"215","DOI":"10.4064\/fm181-3-2","article-title":"On the Leibniz \u2013 Mycielski axiom in set theory","volume":"181","author":"Enayat","year":"2004","journal-title":"Fundam. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.apal.2015.12.002","article-title":"Counterexamples to countable-section \u03a021 uniformization and \u03a031 separation","volume":"167","author":"Kanovei","year":"2016","journal-title":"Ann. Pure Appl. Logic"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"850013","DOI":"10.1142\/S0219061318500137","article-title":"A model of second-order arithmetic satisfying AC but not DC","volume":"19","author":"Friedman","year":"2019","journal-title":"J. Math. Log."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1016\/j.apal.2018.04.006","article-title":"Definable E0 classes at arbitrary projective levels","volume":"169","author":"Kanovei","year":"2018","journal-title":"Ann. Pure Appl. Log."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1017\/jsl.2018.77","article-title":"Definable minimal collapse functions at arbitrary projective levels","volume":"84","author":"Kanovei","year":"2019","journal-title":"J. Symb. Log."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Kanovei, V., and Lyubetsky, V. (2022). A model in which the separation principle holds for a given effective projective sigma-class. Axioms, 11.","DOI":"10.3390\/axioms11030122"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF02025889","article-title":"Surrealist landscape with figures (a survey of recent results in set theory)","volume":"10","author":"Mathias","year":"1979","journal-title":"Period. Math. Hung."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0003-4843(77)90004-3","article-title":"Long projective wellorderings","volume":"12","author":"Harrington","year":"1977","journal-title":"Ann. Math. Log."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/S0049-237X(08)71932-3","article-title":"Some applications of almost disjoint sets","volume":"Volume 59","year":"1970","journal-title":"Studies in Logic and the Foundations of Mathematics"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"279","DOI":"10.4064\/fm-81-4-279-290","article-title":"A new construction of a non-constructible \u039431 subset of \u03c9","volume":"81","author":"Jensen","year":"1974","journal-title":"Fundam. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/354\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:54:48Z","timestamp":1760140488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/354"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,22]]},"references-count":30,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080354"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080354","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,7,22]]}}}