{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T07:58:59Z","timestamp":1764403139854,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,10]],"date-time":"2022-03-10T00:00:00Z","timestamp":1646870400000},"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","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>In this paper, we prove the following: If n\u22653, there is a generic extension of L\u2014the constructible universe\u2014in which it is true that the Separation principle holds for both effective (lightface) classes \u03a3n1 and \u03a0n1 of sets of integers. The result was announced long ago by Leo Harrington with a sketch of the proof for n=3; its full proof has never been presented. Our methods are based on a countable product of almost-disjoint forcing notions independent in the sense of Jensen\u2013Solovay.<\/jats:p>","DOI":"10.3390\/axioms11030122","type":"journal-article","created":{"date-parts":[[2022,3,10]],"date-time":"2022-03-10T11:46:47Z","timestamp":1646912807000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Model in Which the Separation Principle Holds for a Given Effective Projective Sigma-Class"],"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":[[2022,3,10]]},"reference":[{"key":"ref_1","unstructured":"Lusin, N. 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