{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:59:54Z","timestamp":1777478394197,"version":"3.51.4"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T00:00:00Z","timestamp":1693526400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Conway\u2019s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway\u2019s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally.<\/jats:p>\n                  <jats:p>This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.<\/jats:p>","DOI":"10.2478\/forma-2023-0020","type":"journal-article","created":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T13:42:19Z","timestamp":1704116539000},"page":"215-228","source":"Crossref","is-referenced-by-count":3,"title":["The Ring of Conway Numbers in Mizar"],"prefix":"10.2478","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7099-1669","authenticated-orcid":false,"given":"Karol","family":"P\u0105k","sequence":"first","affiliation":[{"name":"Faculty of Computer Science , University of Bia\u0142ystok , Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,12,31]]},"reference":[{"key":"2026042801415205544_j_forma-2023-0020_ref_001","unstructured":"Maan T. Alabdullah, Essam El-Seidy, and Neveen S. Morcos. On numbers and games. International Journal of Scientific and Engineering Research, 11:510\u2013517, February 2020."},{"key":"2026042801415205544_j_forma-2023-0020_ref_002","unstructured":"Norman L. Alling. Foundations of Analysis Over Surreal Number Fields. Number 141 in Annals of Discrete Mathematics. North-Holland, 1987. ISBN 9780444702265."},{"key":"2026042801415205544_j_forma-2023-0020_ref_003","unstructured":"Heinz Bachmann. Transfinite Zahlen. Ergebnisse der Mathematik und ihrer Grenzgebiete, (1). Springer, Berlin, 2., neubearb. aufl. edition, 1967."},{"key":"2026042801415205544_j_forma-2023-0020_ref_004","doi-asserted-by":"crossref","unstructured":"Chad E. Brown and Karol P\u0105k. A tale of two set theories. In Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, and Claudio Sacerdoti Coen, editors, Intelligent Computer Mathematics \u2013 12th International Conference, CICM 2019, CIIRC, Prague, Czech Republic, July 8-12, 2019, Proceedings, volume 11617 of Lecture Notes in Computer Science, pages 44\u201360. Springer, 2019. doi:10.1007\/978-3-030-23250-4_4.","DOI":"10.1007\/978-3-030-23250-4_4"},{"key":"2026042801415205544_j_forma-2023-0020_ref_005","unstructured":"John Horton Conway. On Numbers and Games. A K Peters Ltd., Natick, MA, second edition, 2001. ISBN 1-56881-127-6."},{"key":"2026042801415205544_j_forma-2023-0020_ref_006","unstructured":"Oliver Deiser. Einf\u00fchrung in die Mengenlehre: die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo. Springer, Berlin, 2., verb. und erw. aufl. edition, 2004. ISBN 3-540-20401-6."},{"key":"2026042801415205544_j_forma-2023-0020_ref_007","doi-asserted-by":"crossref","unstructured":"Sebastian Koch. Natural addition of ordinals. Formalized Mathematics, 27(2):139\u2013152, 2019. doi:10.2478\/forma-2019-0015.","DOI":"10.2478\/forma-2019-0015"},{"key":"2026042801415205544_j_forma-2023-0020_ref_008","doi-asserted-by":"crossref","unstructured":"Karol P\u0105k. Conway numbers \u2013 formal introduction. Formalized Mathematics, 31(1): 193\u2013203, 2023. doi:10.2478\/forma-2023-0018.","DOI":"10.2478\/forma-2023-0018"},{"key":"2026042801415205544_j_forma-2023-0020_ref_009","doi-asserted-by":"crossref","unstructured":"Karol P\u0105k. Integration of game theoretic and tree theoretic approaches to Conway numbers. Formalized Mathematics, 31(1):205\u2013213, 2023. doi:10.2478\/forma-2023-0019.","DOI":"10.2478\/forma-2023-0019"},{"key":"2026042801415205544_j_forma-2023-0020_ref_010","doi-asserted-by":"crossref","unstructured":"Dierk Schleicher and Michael Stoll. An introduction to Conway\u2019s games and numbers. Moscow Mathematical Journal, 6:359\u2013388, 2006. doi:10.17323\/1609-4514-2006-6-2-359-388.","DOI":"10.17323\/1609-4514-2006-6-2-359-388"}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2023-0020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:02:30Z","timestamp":1777384950000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2023-0020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,1]]},"references-count":10,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,12,31]]},"published-print":{"date-parts":[[2023,9,1]]}},"alternative-id":["10.2478\/forma-2023-0020"],"URL":"https:\/\/doi.org\/10.2478\/forma-2023-0020","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,1]]}}}