{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T18:42:18Z","timestamp":1757616138146,"version":"3.44.0"},"publisher-location":"Cham","reference-count":14,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783031087394"},{"type":"electronic","value":"9783031087400"}],"license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"DOI":"10.1007\/978-3-031-08740-0_6","type":"book-chapter","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T14:33:51Z","timestamp":1656167631000},"page":"64-73","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lower Bounds on $$\\beta (\\alpha )$$"],"prefix":"10.1007","author":[{"given":"Merlin","family":"Carl","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,26]]},"reference":[{"key":"6_CR1","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1112\/jlms\/s2-14.3.387","volume":"2","author":"FG Abramson","year":"1976","unstructured":"Abramson, F.G., Sacks, G.E.: Uncountable gandy ordinals. J. London Math. Soc.-Second Ser. 2, 387\u2013392 (1976)","journal-title":"J. London Math. Soc.-Second Ser."},{"key":"6_CR2","doi-asserted-by":"crossref","unstructured":"Buchholz, W.: Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies (1981)","DOI":"10.1007\/BFb0091894"},{"key":"6_CR3","doi-asserted-by":"crossref","unstructured":"Carl, M.: Ordinal Computability: An Introduction to Infinitary Machines. De Gruyter (2019)","DOI":"10.1515\/9783110496154"},{"key":"6_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2021.103041","volume":"173","author":"M Carl","year":"2022","unstructured":"Carl, M.: Taming Koepke\u2019s Zoo II: register machines. Ann. Pure Appl. Log. 173, 103041 (2022)","journal-title":"Ann. Pure Appl. Log."},{"key":"6_CR5","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s00153-009-0167-x","volume":"49","author":"M Carl","year":"2010","unstructured":"Carl, M., Fischbach, T., Koepke, P., Miller, R.G., Nasfi, M., Weckbecker, G.: The basic theory of infinite time register machines. Arch. Math. Logic 49, 249\u2013273 (2010)","journal-title":"Arch. Math. Logic"},{"key":"6_CR6","doi-asserted-by":"crossref","unstructured":"Carl, M., Galeotti, L.: Resetting infinite time Blum-Shub-Smale-machines. arXiv Logic (2020)","DOI":"10.1007\/978-3-030-80049-9_7"},{"key":"6_CR7","doi-asserted-by":"crossref","unstructured":"Cutland, N.J.: Computability: An Introduction to Recursive Function Theory (1980)","DOI":"10.1017\/CBO9781139171496"},{"key":"6_CR8","doi-asserted-by":"crossref","unstructured":"Gitman, V., Hamkins, J.D., Johnstone, T.A.: What is the theory ZFC without power set? Math. Log. Q. 62, 391\u2013406 (2016)","DOI":"10.1002\/malq.201500019"},{"key":"6_CR9","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/0003-4843(79)90025-1","volume":"17","author":"R Gostanian","year":"1979","unstructured":"Gostanian, R.: The next admissible ordinal. Ann. Math. Log. 17, 171\u2013203 (1979)","journal-title":"Ann. Math. Log."},{"key":"6_CR10","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/S0049-237X(08)71880-9","volume":"107","author":"G J\u00e4ger","year":"1982","unstructured":"J\u00e4ger, G.: Iterating admissibility in proof theory. Stud. Log. Found. Math. 107, 137\u2013146 (1982)","journal-title":"Stud. Log. Found. Math."},{"key":"6_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1007\/978-3-642-03073-4_29","volume-title":"Mathematical Theory and Computational Practice","author":"P Koepke","year":"2009","unstructured":"Koepke, P.: Ordinal computability. In: Ambos-Spies, K., L\u00f6we, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 280\u2013289. Springer, Heidelberg (2009). https:\/\/doi.org\/10.1007\/978-3-642-03073-4_29"},{"key":"6_CR12","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10469-017-9425-x","volume":"56","author":"P Koepke","year":"2017","unstructured":"Koepke, P., Morozov, A.S.: The computational power of infinite time Blum-Shub-Smale machines. Algebra Log. 56, 37\u201362 (2017)","journal-title":"Algebra Log."},{"key":"6_CR13","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/s00153-008-0093-3","volume":"47","author":"P Koepke","year":"2008","unstructured":"Koepke, P., Siders, R.C.: Register computations on ordinals. Arch. Math. Log. 47, 529\u2013548 (2008)","journal-title":"Arch. Math. Log."},{"key":"6_CR14","doi-asserted-by":"crossref","unstructured":"Sacks, G.E.: Higher recursion theory (1990)","DOI":"10.1007\/978-3-662-12013-2"}],"container-title":["Lecture Notes in Computer Science","Revolutions and Revelations in Computability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-08740-0_6","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T17:14:14Z","timestamp":1757092454000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-08740-0_6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"ISBN":["9783031087394","9783031087400"],"references-count":14,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-08740-0_6","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2022]]},"assertion":[{"value":"26 June 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"CiE","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Conference on Computability in Europe","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Swansea","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"United Kingdom","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2022","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"11 July 2022","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"15 July 2022","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"18","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"cie2022","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/cs.swansea.ac.uk\/cie2022\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}