{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,28]],"date-time":"2026-06-28T09:47:12Z","timestamp":1782640032610,"version":"3.54.5"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T00:00:00Z","timestamp":1706572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Russian Science Foundation","award":["24-44-00099"],"award-info":[{"award-number":["24-44-00099"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>It was established by Jensen in 1970 that there is a generic extension L[a] of the constructible universe L by a non-constructible real a\u2209L, minimal over L, such that a is \u039431 in L[a]. Our first main theorem generalizes Jensen\u2019s result by constructing, for each n\u22652, a generic extension L[a] by a non-constructible real a\u2209L, still minimal over L, such that a is \u0394n+11 in L[a] but all \u03a3n1 reals are constructible in L[a]. Jensen\u2019s forcing construction has found a number of applications in modern set theory. A problem was recently discussed as to whether Jensen\u2019s construction can be reproduced entirely by means of second-order Peano arithmetic PA2, or, equivalently, ZFC\u2212 (minus the power set axiom). The obstacle is that the proof of the key CCC property (whether by Jensen\u2019s original argument or the later proof using the diamond technique) essentially involves countable elementary submodels of L\u03c92, which is way beyond ZFC\u2212. We demonstrate how to circumvent this difficulty by means of killing only definable antichains in the course of a Jensen-like transfinite construction of the forcing notion, and then use this modification to define a model with a minimal \u0394n+11 real as required as a class-forcing extension of a model of ZFC\u2212 plus V=L.<\/jats:p>","DOI":"10.3390\/axioms13020096","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T12:06:58Z","timestamp":1706616418000},"page":"96","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Jensen \u0394n1 Reals by Means of ZFC and Second-Order Peano Arithmetic"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7415-9784","authenticated-orcid":false,"given":"Vladimir","family":"Kanovei","sequence":"first","affiliation":[{"name":"Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow 127051, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3739-9161","authenticated-orcid":false,"given":"Vassily","family":"Lyubetsky","sequence":"additional","affiliation":[{"name":"Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow 127051, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"ref_1","unstructured":"Bar-Hillel, Y. (1962). Essays on the Foundations of Mathematics, Dedicated to A. A. Fraenkel on His 70th Anniversary, North-Holland."},{"key":"ref_2","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_3","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_4","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/S0049-237X(08)71932-3","article-title":"Some applications of almost disjoint sets","volume":"59","author":"Jensen","year":"1970","journal-title":"Stud. Log. Found. Math."},{"key":"ref_5","first-page":"50","article-title":"A nonconstructible \u039431 set of integers","volume":"127","author":"Solovay","year":"1967","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/0001-8708(85)90005-2","article-title":"Minimal model of \u201c\n\t\t  \u21351L is countable\u201d and definable reals","volume":"55","author":"Abraham","year":"1985","journal-title":"Adv. Math."},{"key":"ref_7","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. Logic"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0003-4843(82)90002-X","article-title":"\u039431 31 reals","volume":"23","author":"David","year":"1982","journal-title":"Ann. Math. Logic"},{"key":"ref_9","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."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kanovei, V., and Lyubetsky, V. (2020). Models of set theory in which nonconstructible reals first appear at a given projective level. Mathematics, 8.","DOI":"10.3390\/math8060910"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/0003-4843(74)90001-1","article-title":"Second order arithmetic and related topics","volume":"6","author":"Apt","year":"1974","journal-title":"Ann. Math. Logic"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"321","DOI":"10.2307\/2270324","article-title":"A survey of proof theory","volume":"33","author":"Kreisel","year":"1968","journal-title":"J. Symb. Log."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Simpson, S.G. (2009). Subsystems of Second Order Arithmetic, Cambridge University Press. [2nd ed.]. Perspectives in Logic.","DOI":"10.1017\/CBO9780511581007"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s10992-014-9327-5","article-title":"A disquotational theory of truth as strong as Z2\u2212","volume":"44","author":"Schindler","year":"2015","journal-title":"J. Philos. Log."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/0001-8708(81)90021-9","article-title":"On the necessary use of abstract set theory","volume":"41","author":"Friedman","year":"1981","journal-title":"Adv. Math."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"131","DOI":"10.4064\/fm-93-2-131-144","article-title":"On weaker forms of choice in second order arithmetic","volume":"93","author":"Guzicki","year":"1976","journal-title":"Fundam. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1002\/malq.201500019","article-title":"What is the theory ZFC without power set?","volume":"62","author":"Gitman","year":"2016","journal-title":"Math. Log. Q."},{"key":"ref_19","unstructured":"Daghighi, A., Rezus, A., Pourmahdian, A., Gabbay, D., and Fitting, M. Research Trends in Contemporary Logic, Available online: https:\/\/philpapers.org\/go.pl?aid=ANTMCF."},{"key":"ref_20","unstructured":"Gitman, V., and Matthews, R. (2023). ZFC without power set II: Reflection strikes back. arXiv."},{"key":"ref_21","unstructured":"Jech, T. (2003). Set Theory, Springer. The third millennium revised and expanded ed."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm725-9-2020","article-title":"Kelley-Morse set theory does not prove the class Fodor principle","volume":"254","author":"Gitman","year":"2021","journal-title":"Fundam. Math."},{"key":"ref_23","unstructured":"Harrington, L. (2024, January 15). The Constructible Reals Can Be Anything. Preprint Dated May 1974 with Several Addenda Dated up to October 1975. Available online: http:\/\/logic-library.berkeley.edu\/catalog\/detail\/2135."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0003-4843(79)90010-X","article-title":"Iterated perfect-set forcing","volume":"17","author":"Baumgartner","year":"1979","journal-title":"Ann. Math. Logic"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1017\/jsl.2017.75","article-title":"ZFC proves that the class of ordinals is not weakly compact for definable classes","volume":"83","author":"Enayat","year":"2018","journal-title":"J. Symb. Log."},{"key":"ref_26","unstructured":"Friedman, S.D. (2000). De Gruyter Series in Logic and Its Applications, de Gruyter."},{"key":"ref_27","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_28","first-page":"657","article-title":"A minimal model for \u00acCH: Iteration of Jensen\u2019s reals","volume":"281","author":"Abraham","year":"1984","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1850013","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_30","unstructured":"Wansner, L., and Wontner, N.J.H. (2023). Descriptive Choice Principles and How to Separate Them. arXiv."},{"key":"ref_31","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_32","doi-asserted-by":"crossref","unstructured":"Kanovei, V., and Lyubetsky, V. (2022). A model in which wellorderings of the reals appear at a given projective level. Axioms, 11.","DOI":"10.3390\/axioms11080354"},{"key":"ref_33","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. Logic"},{"key":"ref_34","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. Logic"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.2307\/2275646","article-title":"An Ulm-type classification theorem for equivalence relations in Solovay model","volume":"62","author":"Kanovei","year":"1997","journal-title":"J. Symbolic Logic"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/96\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:51:57Z","timestamp":1760104317000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/2\/96"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,30]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["axioms13020096"],"URL":"https:\/\/doi.org\/10.3390\/axioms13020096","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,30]]}}}