{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T16:31:04Z","timestamp":1703089864859},"reference-count":17,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":3603,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1997,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We first note that Gentzen's proof\u2010reduction for his consistency proof of PA can be directly interpreted as moves of Kirby\u2010Paris' Hydra Game, which implies a direct independence proof of the game (Section 1 and Appendix). Buchholz's Hydra Game for labeled hydras is known to be much stronger than PA. However, we show that the one\u2010dimensional version of Buchholz's Game can be exactly identified to Kirby\u2010Paris' Game (which is two\u2010dimensional but without labels), by a simple and natural interpretation (Section 2). Jervell proposed another type of a combinatorial game, by abstracting Gentzen's proof\u2010reductions and showed that his game is independent of PA. We show (Section 3) that this Jervell's game is actually much stronger than PA, by showing that the critical ordinal of Jervell's game is \u03c6<jats:sub>\u03c9<\/jats:sub> (0) (while that of PA or of Kirby\u2010Paris' Game is \u03c6<jats:sub>1<\/jats:sub> (0) = \u03f5<jats:sub>0<\/jats:sub>) in the Veblen hierarchy of ordinals.<\/jats:p>","DOI":"10.1002\/malq.19970430113","type":"journal-article","created":{"date-parts":[[2007,5,30]],"date-time":"2007-05-30T15:24:18Z","timestamp":1180538658000},"page":"103-120","source":"Crossref","is-referenced-by-count":11,"title":["A Relationship Among Gentzen's Proof\u2010Reduction, Kirby\u2010Paris' Hydra Game and Buchholz's Hydra Game"],"prefix":"10.1002","volume":"43","author":[{"given":"Masahiro","family":"Hamano","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mitsuhiro","family":"Okada","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01175640"},{"key":"e_1_2_1_3_2","doi-asserted-by":"crossref","first-page":"21","DOI":"10.21099\/tkbjm\/1496160191","article-title":"A subsystem of classical analysis proper to reduction method for \u03c01\n                  1\u2010analysis","volume":"9","author":"Arai T.","year":"1985","journal-title":"Tsukuba J. Math."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90078-9"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19940400212"},{"key":"e_1_2_1_6_2","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1090\/S0002-9939-1983-0687646-0","article-title":"A short proof of two recently discovered independence results using recursion theoretic methods","volume":"87","author":"Cichon E. A.","year":"1983","journal-title":"Proc. Amer. Math. Soc."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.2307\/2269866"},{"key":"e_1_2_1_8_2","first-page":"19","article-title":"Neue Fassung des Widerspruchsfreiheitbeweises f\u00fcr die reine Zahlentheorie","volume":"4","author":"Gentzen G.","year":"1938","journal-title":"Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, N.F."},{"key":"e_1_2_1_9_2","first-page":"105","article-title":"D\u00e9monstration de l'\u03c9\u2010non\u2010contradiction de l'arithem\u00e9tique","volume":"3","author":"Hanatani Y.","year":"1968","journal-title":"Ann. Japan Assoc. Philos. Sci."},{"key":"e_1_2_1_10_2","article-title":"A direct independence proof of Buchholz's hydra game on finite labeled trees","author":"Hamano M.","journal-title":"Archive Math. Logic"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19850312506"},{"key":"e_1_2_1_12_2","doi-asserted-by":"crossref","unstructured":"Gardner M. Mathematical games (tasks you cannot help finishing no matter how hard you try to block finishing them). Scientific American. August (1983) 12\u201321.","DOI":"10.1038\/scientificamerican0883-12"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/14.4.285"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.2307\/2274350"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(88)80016-7"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66473-1"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007558"},{"key":"e_1_2_1_18_2","volume-title":"Proof Theory","author":"Takeuti G.","year":"1987"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19970430113","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19970430113","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T10:38:58Z","timestamp":1698489538000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19970430113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,1]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1997,1]]}},"alternative-id":["10.1002\/malq.19970430113"],"URL":"https:\/\/doi.org\/10.1002\/malq.19970430113","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,1]]}}}