{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T04:43:36Z","timestamp":1772426616005,"version":"3.50.1"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T00:00:00Z","timestamp":1712534400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n In the study of the arithmetic degrees the\n \n \n \n $\\omega \\text {-REA}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the\n \n \n \n $\\omega \\text {-REA}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of an\n \n \n \n $\\omega \\text {-REA}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e. sets fail to show that no\n \n \n \n $\\omega \\text {-REA}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n set is arithmetically minimal. Finally, it constructs a\n \n \n \n $\\prod ^0_{2}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n singleton of minimal arithmetic degree. Not only is this a result of considerable interest in its own right, constructions of\n \n \n \n $\\prod ^0_{2}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n singletons often pave the way for constructions of\n \n \n \n $\\omega \\text {-REA}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.\n <\/jats:p>","DOI":"10.1017\/jsl.2024.23","type":"journal-article","created":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T05:15:30Z","timestamp":1712553330000},"page":"336-368","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A \n$ \\prod _{2}^{0}$\n SINGLETON OF MINIMAL ARITHMETIC DEGREE"],"prefix":"10.1017","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8330-6926","authenticated-orcid":false,"given":"PETER M.","family":"GERDES","sequence":"first","affiliation":[{"name":"INDIANA UNIVERSITY BLOOMINGTON"}]}],"member":"56","published-online":{"date-parts":[[2024,4,8]]},"reference":[{"key":"S0022481224000239_r11","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0276079"},{"key":"S0022481224000239_r13","doi-asserted-by":"publisher","DOI":"10.2307\/1970028"},{"key":"S0022481224000239_r6","doi-asserted-by":"crossref","unstructured":"[6] Jockusch, C. G. and Shore, R. A. , Pseudo-jump operators. II: Transfinite iterations, hierarchies and minimal covers, Journal of Symbolic Logic, vol. 49 (1984), no. 4, pp. 1205\u20131236.","DOI":"10.2307\/2274273"},{"key":"S0022481224000239_r7","volume-title":"Randomness and genericity in the degrees of unsolvability","author":"Kurtz","year":"1981"},{"key":"S0022481224000239_r2","doi-asserted-by":"crossref","unstructured":"[2] Friedman, H. , One hundred and two problems in mathematical logic, Journal of Symbolic Logic, vol. 40 (1975), no. 2, pp. 113\u2013129.","DOI":"10.2307\/2271891"},{"key":"S0022481224000239_r8","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-16.1.537"},{"key":"S0022481224000239_r14","unstructured":"[14] Simpson, M. F. , Arithmetic degrees: Initial segments, $\\boldsymbol{\\omega}$ -rea operators and the $\\boldsymbol{\\omega}$ -jump, Ph.D. thesis , Cornell University, 1985."},{"key":"S0022481224000239_r4","unstructured":"[4] Harrington, L. A. , Arithmetically incomparable arithmetic singletons, unpublished mimeographed notes, 1976."},{"key":"S0022481224000239_r1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2013.10.004"},{"key":"S0022481224000239_r9","volume-title":"Classical Recursion Theory","author":"Odifreddi","year":"1992"},{"key":"S0022481224000239_r10","doi-asserted-by":"publisher","DOI":"10.2307\/1970393"},{"key":"S0022481224000239_r15","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-3507386"},{"key":"S0022481224000239_r12","doi-asserted-by":"publisher","DOI":"10.1017\/9781316717301"},{"key":"S0022481224000239_r3","unstructured":"[3] Gerdes, P. M. , Harrington\u2019s solution to McLaughlin\u2019s conjecture and non-uniform self-moduli, unpublished preprint, 2010. https:\/\/arxiv.org\/abs\/1012.3427."},{"key":"S0022481224000239_r5","unstructured":"[5] Harrington, L. A. , Mclaughlin\u2019s conjecture, unpublished handwritten notes, 1976."}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481224000239","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T02:28:38Z","timestamp":1772418518000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481224000239\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,8]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0022481224000239"],"URL":"https:\/\/doi.org\/10.1017\/jsl.2024.23","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,8]]},"assertion":[{"value":"\u00a9 The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}