{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,13]],"date-time":"2024-09-13T18:07:02Z","timestamp":1726250822176},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>By [12] we know that transfinite induction up to \u0398<jats:sub>\u03b5\u03a9<jats:italic>N<\/jats:italic>+1<\/jats:sub>0 is not provable in ID<jats:sub><jats:italic>N<\/jats:italic><\/jats:sub>, the theory of <jats:italic>N<\/jats:italic>-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than \u0398<jats:sub>\u03b5\u03a9<jats:italic>N<\/jats:italic>+1<\/jats:sub>0 is provable in ID<jats:sub arrange=\"stack\">N<\/jats:sub><jats:sup arrange=\"stack\">i<\/jats:sup>, the intuitionistic version of ID<jats:sub><jats:italic>N<\/jats:italic><\/jats:sub>, and extend this result to theories for transfinitely iterated inductive definitions.<\/jats:p><jats:p>In [14] Sch\u00fctte proves the wellordering of his notational systems using predicates <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_inline1\" \/> is wellordered) with <jats:italic>M<\/jats:italic><jats:sub>\u03ba<\/jats:sub> \u2254 {<jats:italic>x<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_inline1\" \/> and 0 \u2264 \u03ba \u2264 <jats:italic>N<\/jats:italic>. Obviously the predicates <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_inline3\" \/> are definable in ID<jats:sub arrange=\"stack\">N<\/jats:sub><jats:sup arrange=\"stack\">i<\/jats:sup> with the defining axioms:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where Prog [<jats:italic>M<\/jats:italic><jats:sub>\u03ba<\/jats:sub>, <jats:italic>X<\/jats:italic>] means that <jats:italic>X<\/jats:italic> is progressive with respect to <jats:italic>M<\/jats:italic><jats:sub>\u03ba<\/jats:sub>, i.e.<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>The crucial point in Sch\u00fctte's wellordering proof is Lemma 19 [14, p. 130] which can be modified to<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120004994X_equ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>where TI[<jats:italic>M<\/jats:italic><jats:sub>\u03ba + 1<\/jats:sub>, <jats:italic>a<\/jats:italic>] is the scheme of transfinite induction over <jats:italic>M<\/jats:italic><jats:sub>\u03ba + 1<\/jats:sub> up to <jats:italic>a<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2271954","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"118-125","source":"Crossref","is-referenced-by-count":20,"title":["Provable wellorderings of formal theories for transfinitely iterated inductive definitions"],"prefix":"10.1017","volume":"43","author":[{"given":"W.","family":"Buchholz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"W.","family":"Pohlers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004994X_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066745"},{"key":"S002248120004994X_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66473-1"},{"key":"S002248120004994X_ref011","volume-title":"Communications of the Mathematics Institute of Rijksuniversiteit","author":"Pfeiffer","year":"1973"},{"key":"S002248120004994X_ref006","first-page":"327","volume-title":"Intuitionism and proof theory","author":"Gerber","year":"1970"},{"key":"S002248120004994X_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF01564760"},{"key":"S002248120004994X_ref004","first-page":"303","volume-title":"Intuitionism and proof theory","author":"Feferman","year":"1970"},{"key":"S002248120004994X_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02276806"},{"key":"S002248120004994X_ref002","first-page":"4","volume-title":"Proof Theory Symposium, Kiel, 1974, Lecture Notes in Mathematics","author":"Buchholz","year":"1975"},{"key":"S002248120004994X_ref001","first-page":"171","volume":"40","author":"Bridge","year":"1975","journal-title":"A simplification of the Bachmann method for generating large countable ordinals"},{"key":"S002248120004994X_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF01967653"},{"key":"S002248120004994X_ref012","first-page":"271","volume-title":"Proof Theory Symposium, Kiel, 1974, Lecture Notes in Mathematics","author":"Pohlers","year":"1975"},{"key":"S002248120004994X_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01982045"},{"key":"S002248120004994X_ref007","first-page":"355","volume":"37","author":"Howard","year":"1972","journal-title":"A system of abstract constructive ordinals"},{"key":"S002248120004994X_ref013","unstructured":"Pohlers W. , Ordinals connected with formal theories of transfinitely iterated inductive definitions, this Journal, (to appear)."},{"key":"S002248120004994X_ref008","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/01340346"},{"key":"S002248120004994X_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01967820"}],"container-title":["Journal of Symbolic 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