{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,25]],"date-time":"2024-07-25T16:03:22Z","timestamp":1721923402092},"reference-count":109,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[1987,12,1]],"date-time":"1987-12-01T00:00:00Z","timestamp":565315200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Arch math Logik"],"published-print":{"date-parts":[[1987,12]]},"DOI":"10.1007\/bf02017491","type":"journal-article","created":{"date-parts":[[2005,8,12]],"date-time":"2005-08-12T07:11:20Z","timestamp":1123830680000},"page":"57-76","source":"Crossref","is-referenced-by-count":11,"title":["Natural well-orderings"],"prefix":"10.1007","volume":"26","author":[{"given":"John N.","family":"Crossley","sequence":"first","affiliation":[]},{"given":"Jane Bridge","family":"Kister","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"BF02017491_CR1","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/BF01175640","volume":"53","author":"W. Ackermann","year":"1950","unstructured":"W. Ackermann [1950] Konstruktiver Aufbau eines Abschnitts der zweiten Cantorschen Zahlenklasse. Math. Zeit.53 (1950) 403\u2013413.","journal-title":"Math. Zeit."},{"key":"BF02017491_CR2","unstructured":"P. H. G. Aczel [1966] Mathematical problems in logic. D. Phil. thesis, Oxford."},{"key":"BF02017491_CR3","doi-asserted-by":"crossref","first-page":"430","DOI":"10.2307\/2270848","volume":"32","author":"P. H. G. Aczel","year":"1967","unstructured":"[1967] Normal functors on linear orderings. (Abstract) J. Symbolic Logic32 (1967) 430.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR4","unstructured":"[1969] A new approach to the Bachmann method for describing countable ordinals. (Preliminary summary) (Unpublished)."},{"key":"BF02017491_CR5","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2307\/2272543","volume":"37","author":"P. H. G. Aczel","year":"1972","unstructured":"[1972] Describing ordinals using functionals of transfinite type. J. Symbolic Logic37 (1972) 35\u201347.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR6","first-page":"115","volume":"95","author":"H. Bachmann","year":"1950","unstructured":"H. Bachmann [1950] Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen. Vierteljahresschr. Naturforsch. Ges. Z\u00fcrich95 (1950) 115\u2013147.","journal-title":"Vierteljahresschr. Naturforsch. Ges. Z\u00fcrich"},{"key":"BF02017491_CR7","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF02564291","volume":"26","author":"H. Bachmann","year":"1952","unstructured":"[1952] Vergleich und Kombination zweier Methoden von Veblen und Finsler zur L\u00f6sung des Problems der ausgezeichneten Folgen von Ordnungszahlen. Comment. Math. Helv.26 (1952) 55\u201367.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR8","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/BF02566922","volume":"28","author":"H. Bachmann","year":"1954","unstructured":"[1954] Normalfunktionen und Hauptfolgen. Comment. Math. Helv.28 (1954) 9\u201316.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR9","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-52756-2","volume-title":"Transfinite Zahlen","author":"H. Bachmann","year":"1955","unstructured":"[1955]Transfinite Zahlen. Springer, Berlin."},{"key":"BF02017491_CR10","unstructured":"J. E. Bridge [1972] Some problems in mathematical logic. Systems of ordinal functions and ordinal notations. D. Phil. thesis, Oxford."},{"key":"BF02017491_CR11","doi-asserted-by":"crossref","first-page":"171","DOI":"10.2307\/2271898","volume":"40","author":"J. E. Bridge","year":"1975","unstructured":"[1975] A simplification of the Bachmann method for generating large countable ordinals. J. Symbolic Logic40 (1975) 171\u2013185.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR12","unstructured":"[1978] A summary of the literature concerning ordinal notations. (Unpublished)"},{"key":"BF02017491_CR13","unstructured":"L. E. J. Brouwer [1918\u201319] Begr\u00fcndung der Mengenlehre unabh\u00e4ngig vom logischen Satz vom ausgeschlossenen Dritten. Reprinted inCollected works (Ed: A. Heyting, Vol. I [150]\u2013[221]."},{"key":"BF02017491_CR14","unstructured":"W. Buchholz [1974] Rekursive Bezeichnungssysteme f\u00fcr Ordinalzahlen auf der Grundlage der Feferman-Aczelschen Normalfunktionen\u0398 \u03b1. Dissertation, Munich."},{"key":"BF02017491_CR15","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/BFb0079544","volume-title":"Proof theory symposium, Kiel 1974","author":"W. Buchholz","year":"1975","unstructured":"[1975] Normalfunktionen und konstruktive Systeme von Ordinalzahlen.Proof theory symposium, Kiel 1974, pp. 4\u201325, Lecture Notes in Mathematics, 500, Springer, Berlin."},{"key":"BF02017491_CR16","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF02007261","volume":"18","author":"W. Buchholz","year":"1976","unstructured":"[1976] \u00dcber Teilsysteme von $$\\bar \\theta (\\{ g\\} )$$ . Arch. Math. Logik Grundlag.18 (1976) 85\u201398.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR17","unstructured":"[1982] Collapsing functions. (Preprint)"},{"key":"BF02017491_CR18","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0168-0072(86)90052-7","volume":"32","author":"W. Buchholz","year":"1986","unstructured":"[1986] A new system of proof theoretic ordinal functions. Ann. Pure Appl. Logic32 (1986) 195\u2013207.","journal-title":"Ann. Pure Appl. Logic"},{"key":"BF02017491_CR19","unstructured":"[198?] An independence result for (\u03a0 1 1 -CA) + BI. To appear in Ann. Pure Appl. Logic."},{"key":"BF02017491_CR20","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0091894","volume-title":"Iterated inductive definitions and subsystems of analysis: Recent prooftheoretical studies","author":"W. Buchholz","year":"1981","unstructured":"W. Buchholz, S. Feferman, W. Pohlers, and W. Sieg [1981]Iterated inductive definitions and subsystems of analysis: Recent prooftheoretical studies. Lecture Notes in Mathematics, 897, Springer, Berlin."},{"key":"BF02017491_CR21","doi-asserted-by":"crossref","first-page":"118","DOI":"10.2307\/2271954","volume":"43","author":"W. Buchholz","year":"1978","unstructured":"W. Buchholz and W. Pohlers [1978] Provable well-orderings of formal theories for transfinitely iterated inductive definitions. J. Symbolic Logic43 (1978) 118\u2013125.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR22","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/BF02276806","volume":"17","author":"W. Buchholz","year":"1976","unstructured":"W. Buchholz and K. Sch\u00fctte [1976] Die Beziehungen zwischen den Ordinalzahlsystemen\u03a3 und $$\\bar \\theta (\\omega )$$ . Arch. Math. Logik Grundlag.17 (1976) 179\u2013189.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR23","first-page":"99","volume":"1983","author":"W. Buchholz","year":"1983","unstructured":"[1983] Ein Ordinalzahlensystem f\u00fcr die beweistheoretische Abgrenzung der \u03a0 1 1 -Separation und Bar-Induktion. Bayer. Akad. Wiss. Math.-Nature. Kl.1983, 99\u2013132.","journal-title":"Bayer. Akad. Wiss. Math.-Nature. Kl."},{"key":"BF02017491_CR24","first-page":"178","volume":"29","author":"A. Church","year":"1927","unstructured":"A. Church [1927] Alternatives to Zermelo's assumption. Trans. Amer. Math. Soc.29 (1927) 178\u2013208.","journal-title":"Trans. Amer. Math. Soc."},{"key":"BF02017491_CR25","doi-asserted-by":"crossref","first-page":"11","DOI":"10.4064\/fm-28-1-11-21","volume":"28","author":"A. Church","year":"1937","unstructured":"A. Church and S. C. Kleene [1937] Formal definitions in the theory of ordinal numbers. Fund. Math.28 (1937) 11\u201321.","journal-title":"Fund. Math."},{"key":"BF02017491_CR26","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2307\/2273557","volume":"48","author":"E. A. Cichon","year":"1983","unstructured":"E. A. Cichon and S. S. Wainer [1983] The slow-growing and Grzegorczyk hierarchies. J. Symbolic Logic48 (1983) 399\u2013408.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0049-237X(08)71691-4","volume-title":"Constructive order types I.Formal systems and recursive functions","author":"J. N. Crossley","year":"1965","unstructured":"J. N. Crossley [1965] Constructive order types I.Formal systems and recursive functions (Eds: J.N. Crossley and M.A.E. Dummett), pp. 189\u2013264, North-Holland, Amsterdam."},{"key":"BF02017491_CR28","volume-title":"Constructive order types","author":"J. N. Crossley","year":"1969","unstructured":"[1969]Constructive order types. North-Holland, Amsterdam."},{"key":"BF02017491_CR29","unstructured":"[1980]The emergence of number. Upside Down A Book Co., Steel's Creek, Victoria, Australia 3775."},{"key":"BF02017491_CR30","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0168-0072(86)90070-9","volume":"31","author":"J. N. Crossley","year":"1986","unstructured":"J. N. Crossley, A. B. M. Manaster, and M. Moses [1986] Recursive categoricity and recursive stability. Ann. Pure Appl. Logic31 (1986) 191\u2013204.","journal-title":"Ann. Pure Appl. Logic"},{"key":"BF02017491_CR31","first-page":"308","volume":"28","author":"J. N. Crossley","year":"1963","unstructured":"J. N. Crossley and R. J. Parikh [1963] On isomorphisms of recursive well-orderings. (Abstract) J. Symbolic Logic28 (1963) 308.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR32","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01973029","volume":"9","author":"J. N. Crossley","year":"1966","unstructured":"J. N. Crossley and K. Sch\u00fctte [1966] Non-uniqueness at \u03c92 in Kleene's \u03d5. Arch. Math. Logik Grundlag.9 (1966) 95\u2013101.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR33","volume-title":"Essays on the theory of numbers","author":"R. Dedekind","year":"1888","unstructured":"R. Dedekind [1888] Was sind und was sollen die Zahlen? English translation in: R. Dedekind,Essays on the theory of numbers (translated by W. Berman), Dover, New York, 1963."},{"key":"BF02017491_CR34","volume-title":"L'\u00e9num\u00e9ration transfinie. 4 volumes","author":"A. Denjoy","year":"1946","unstructured":"A. Denjoy [1946\u201354]L'\u00e9num\u00e9ration transfinie. 4 volumes. Gauthier-Villars, Paris."},{"key":"BF02017491_CR35","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BFb0099482","volume-title":"Computation and proof theory","author":"E. C. Dennis-Jones","year":"1984","unstructured":"E. C. Dennis-Jones and S. S. Wainer [1984] Subrecursive hierarchies via direct limits.Computation and proof theory, pp. 117\u2013128. Lecture Notes in Mathematics, 1104, Springer, Berlin."},{"key":"BF02017491_CR36","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1002\/malq.19700160704","volume":"16","author":"H. C. Doets","year":"1970","unstructured":"H. C. Doets [1970] A generalization in the theory of normal functions. Z. Math. Logik Grundlag. Math.16 (1970) 389\u2013392.","journal-title":"Z. Math. Logik Grundlag. Math."},{"key":"BF02017491_CR37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/2269764","volume":"29","author":"S. Feferman","year":"1964","unstructured":"S. Feferman [1964] Systems of predicative analysis. J. Symbolic Logic29 (1964) 1\u201330.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR38","doi-asserted-by":"crossref","first-page":"193","DOI":"10.2307\/2269866","volume":"33","author":"S. Feferman","year":"1968","unstructured":"[1968] Systems of predicative analysis II: Representations of ordinals. J. Symbolic Logic33 (1968) 193\u2013220.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR39","doi-asserted-by":"crossref","unstructured":"[1970] Hereditarily replete functionals over the ordinals. In Kino et al. [1970], pp. 289\u2013301.","DOI":"10.1016\/S0049-237X(08)70760-2"},{"key":"BF02017491_CR40","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BFb0059538","volume-title":"Conference in mathematical logic \u2014 London '70","author":"S. Feferman","year":"1972","unstructured":"[1972] Infinitary properties, local functors, and systems of ordinals functions.Conference in mathematical logic \u2014 London '70, pp. 63\u201397, Lecture Notes in Mathematics, 255, Springer, Berlin."},{"key":"BF02017491_CR41","unstructured":"[1981] Preface: How we got from there to here. In Buchholz et al. [1981], pp. 1\u201315."},{"key":"BF02017491_CR42","unstructured":"[198?] Proof theory: A personal report. (Preprint)"},{"key":"BF02017491_CR43","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF02566448","volume":"25","author":"P. Finsler","year":"1951","unstructured":"P. Finsler [1951] Eine transfinite Folge arithmetischer Operationen. Comment. Math. Helv.25 (1951) 75\u201390.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR44","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1007\/BF02771107","volume":"5","author":"H. Gaifman","year":"1967","unstructured":"H. Gaifman [1967] A generalization of Mahlo's method for obtaining large cardinal numbers. Israel J. Math.5 (1967) 188\u2013199.","journal-title":"Israel J. Math."},{"key":"BF02017491_CR45","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1016\/S0049-237X(08)71505-2","volume-title":"Sets, models and recursion theory","author":"H. Gaifman","year":"1967","unstructured":"[1967a] Uniform extension operators for models and their applications.Sets, models and recursion theory (Ed: J.N. Crossley), pp. 122\u2013155, North-Holland, Amsterdam."},{"key":"BF02017491_CR46","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF01360719","volume":"174","author":"H. Gerber","year":"1967","unstructured":"H. Gerber [1967] An extension of Sch\u00fctte's Klammersymbols. Math. Ann.174 (1967) 203\u2013216.","journal-title":"Math. Ann."},{"key":"BF02017491_CR47","doi-asserted-by":"crossref","unstructured":"[1970] Brouwer's bar theorem and a system of ordinal notations. In Kino et al. [1970], pp. 327\u2013338.","DOI":"10.1016\/S0049-237X(08)70762-6"},{"key":"BF02017491_CR48","first-page":"59","volume-title":"Colloque International de Logique (Clermont-Ferrand, 1975)","author":"J.-Y. Girard","year":"1977","unstructured":"J.-Y. Girard [1977] Functionals and ordinoids.Colloque International de Logique (Clermont-Ferrand, 1975), pp. 59\u201371, CNRS, Paris."},{"key":"BF02017491_CR49","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/0003-4843(81)90016-4","volume":"21","author":"J.-Y. Girard","year":"1981","unstructured":"[1981]\u03a01\/2-logic. Part I: Dilators. Ann. Math. Logic21 (1981) 75\u2013219.","journal-title":"Ann. Math. Logic"},{"key":"BF02017491_CR50","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0049-237X(09)70184-3","volume-title":"Logic, methodology and philosophy of science VI (Hannover, 1979)","author":"J.-Y. Girard","year":"1982","unstructured":"[1982] A survey of\u03a01\/2-logic.Logic, methodology and philosophy of science VI (Hannover, 1979), pp. 89\u2013107, North-Holland, Amsterdam."},{"key":"BF02017491_CR51","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF00486046","volume":"62","author":"J.-Y. Girard","year":"1985","unstructured":"[1985] Introduction to\u03a01\/2-logic. Synthese62 (1985) 191\u2013216.","journal-title":"Synthese"},{"key":"BF02017491_CR52","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1090\/pspum\/042\/791069","volume":"42","author":"J.-Y. Girard","year":"1985","unstructured":"J.-Y. Girard and J. P. Ressayre [1985] El\u00e9ments de logique\u03a01\/n.Symposia in Pure Math. 42, pp. 389\u2013445, Amer. Math. Soc., Providence, RI.","journal-title":"Symposia in Pure Math."},{"key":"BF02017491_CR53","doi-asserted-by":"crossref","first-page":"713","DOI":"10.2307\/2274127","volume":"49","author":"J.-Y. Girard","year":"1984","unstructured":"J.-Y. Girard and J. Vauzeilles [1984] Functors and ordinal notations I: A functorial construction of the Veblen hierarchy. J. Symbolic Logic49 (1984) 713\u2013729.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR54","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.2307\/2274263","volume":"49","author":"J.-Y. Girard","year":"1984","unstructured":"[1984a] Functors and ordinal notations II: A functorial construction of the Bachmann hierarchy. J. Symbolic Logic49 (1984) 1079\u20131114.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR55","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF02007148","volume":"24","author":"J.-Y. Girard","year":"1984","unstructured":"[1984b] Les premiers recursivement inaccessible et Mahlo et la th\u00e9orie des dilatateurs. Arch. Math. Logik Grundlag.24 (1984) 167\u2013191.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR56","unstructured":"A. Grzegorczyk [1953] Some classes of recursive functions. Rozprawy Mat. No. 4."},{"key":"BF02017491_CR57","first-page":"87","volume":"35","author":"G. H. Hardy","year":"1904","unstructured":"G. H. Hardy [1904] A theorem concerning the infinite cardinal numbers. Quarterly J. Math.35 (1904) 87\u201394.","journal-title":"Quarterly J. Math."},{"key":"BF02017491_CR58","doi-asserted-by":"crossref","first-page":"355","DOI":"10.2307\/2272979","volume":"37","author":"W. A. Howard","year":"1972","unstructured":"W. A. Howard [1972] A system of abstract constructive ordinals. J. Symbolic Logic37 (1972) 355\u2013374.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR59","doi-asserted-by":"crossref","unstructured":"D. Isles [1970] Regular ordinals and normal forms. In Kino et al. [1970], pp. 339\u2013361.","DOI":"10.1016\/S0049-237X(08)70763-8"},{"key":"BF02017491_CR60","doi-asserted-by":"crossref","first-page":"288","DOI":"10.2307\/2270263","volume":"36","author":"D. Isles","year":"1971","unstructured":"[1971] Natural well-orderings. J. Symbolic Logic36 (1971) 288\u2013300.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR61","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF02023014","volume":"23","author":"G. J\u00e4ger","year":"1983","unstructured":"G. J\u00e4ger [1983] A well-ordering proof for Feferman's theoryT 0. Arch. Math. Logik Grundlag.23 (1983) 65\u201377.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR62","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/BF02007140","volume":"24","author":"G. J\u00e4ger","year":"1984","unstructured":"[1984] \u03f1-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlag.24 (1984) 49\u201362.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR63","unstructured":"[1985] Countable admissible ordinals and dilators. (Preprint)"},{"key":"BF02017491_CR64","unstructured":"G. J\u00e4ger and W. Pohlers [1983] Eine beweistheoretische Untersuchung von (\u03941\/2-CA) + (BI) und verwandter Systeme. Bayer. Akad. Wiss. Math.-Natur. Kl.1982, 1\u201328."},{"key":"BF02017491_CR65","volume-title":"Intuitionism and Proof theory","year":"1970","unstructured":"A. Kino, J. Myhill, and R. E. Vesley (Eds.) [1970]Intuitionism and Proof theory. Proceedings of the summer conference. Buffalo, New York, 1968. North-Holland, Amsterdam."},{"key":"BF02017491_CR66","unstructured":"J. E. Kister (see J. E. Bridge)"},{"key":"BF02017491_CR67","doi-asserted-by":"crossref","first-page":"150","DOI":"10.2307\/2267778","volume":"3","author":"S. C. Kleene","year":"1938","unstructured":"S. C. Kleene [1938] On notation for ordinal numbers. J. Symbolic Logic3 (1938) 150\u2013155.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR68","doi-asserted-by":"crossref","first-page":"405","DOI":"10.2307\/2372632","volume":"77","author":"S. C. Kleene","year":"1955","unstructured":"[1955] On the forms of predicates in the theory of constructive ordinals (second paper). Amer. J. Math.77 (1955) 405\u2013428.","journal-title":"Amer. J. Math."},{"key":"BF02017491_CR69","first-page":"95","volume-title":"Lectures in modern mathematics, Vol. III","author":"G. Kreisel","year":"1965","unstructured":"G. Kreisel [1965] Mathematical logic. InLectures in modern mathematics, Vol. III (Ed: T.L. Saaty), pp. 95\u2013195, Wiley, New York."},{"key":"BF02017491_CR70","unstructured":"H. Levitz [1965] On the ordinal notations of Sch\u00fctte and the ordinal diagrams of Takeuti. Ph.D. thesis, Penn. State University."},{"key":"BF02017491_CR71","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/BF02566881","volume":"41","author":"H. Levitz","year":"1966","unstructured":"[1966] \u00dcber die Finslerschen h\u00f6heren arithmetischen Operationen. Comment. Math. Helv.41 (1966) 273\u2013286.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR72","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1002\/malq.19690150708","volume":"15","author":"H. Levitz","year":"1969","unstructured":"[1969] A simplification of Takeuti's ordinal diagrams of finite order. Z. Math. Logik Grundlag. Math.15 (1969) 141\u2013154.","journal-title":"Z. Math. Logik Grundlag. Math."},{"key":"BF02017491_CR73","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF02564515","volume":"44","author":"H. Levitz","year":"1969","unstructured":"[1969a] On the Finsler and Doner-Tarski arithmetical hierarchies. Comment. Math. Helv.44 (1969) 89\u201392.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR74","doi-asserted-by":"crossref","unstructured":"[1970] On the relationship between Takeuti's ordinal diagramsO(n) and Sch\u00fctte's system of notations\u03a3(n). In Kino et al. [1970], pp. 377\u2013405.","DOI":"10.1016\/S0049-237X(08)70765-1"},{"key":"BF02017491_CR75","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1007\/BF02566131","volume":"48","author":"H. Levitz","year":"1973","unstructured":"[1973] A characterization of the Veblen-Sch\u00fctte functions by means of functionals. Comment. Math. Helv.48 (1973) 382\u2013393.","journal-title":"Comment. Math. Helv."},{"key":"BF02017491_CR76","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF01974151","volume":"14","author":"H. Levitz","year":"1971","unstructured":"H. Levitz and K. Sch\u00fctte [1971] A characterization of Takeuti's ordinal diagrams of finite order. Arch. Math. Logik Grundlag.14 (1971) 75\u201397.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR77","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1017\/S0022481200051501","volume":"41","author":"L. W. Miller","year":"1976","unstructured":"L. W. Miller [1976] Normal functions and constructive ordinal notations. J. Symbolic Logic41 (1976) 439\u2013459.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR78","doi-asserted-by":"crossref","first-page":"329","DOI":"10.2140\/pjm.1966.18.329","volume":"18","author":"Y. N. Moschovakis","year":"1966","unstructured":"Y. N. Moschovakis [1966] Many-one degrees of the predicatesH \u03b1 (x). Pacific J. Math.18 (1966) 329\u2013342.","journal-title":"Pacific J. Math."},{"key":"BF02017491_CR79","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/BF01174155","volume":"58","author":"W. Neumer","year":"1953","unstructured":"W. Neumer [1953\u201356] Zur Konstruktion von Ordnungszahlen. Math. Zeit.58 (1953) 391\u2013413; ibid.59 (1953) 434\u2013454; ibid.60 (1954) 1\u201316; ibid.61 (1954) 47\u201369; ibid.64 (1956) 435\u2013456.","journal-title":"Math. Zeit."},{"key":"BF02017491_CR80","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1002\/malq.19570030604","volume":"3","author":"W. Neumer","year":"1957","unstructured":"[1957\u201370] Algorithmen f\u00fcr Ordnungszahlen und Normalfunktionen. Z. Math. Logik Grundlag. Math.3 (1957) 108\u2013150; ibid.6 (1960) 1\u201365; ibid.16 (1970) 1\u2013112.","journal-title":"Z. Math. Logik Grundlag. Math."},{"key":"BF02017491_CR81","volume-title":"Ptykes in G\u00f6del's T und Verallgemeinerte Rekursion \u00fcber Mengen und Ordinalzahlen","author":"P. P\u00e4ppinghaus","year":"1985","unstructured":"P. P\u00e4ppinghaus [1985] Ptykes in G\u00f6del's T und Verallgemeinerte Rekursion \u00fcber Mengen und Ordinalzahlen. Habilitationsschrift, Hannover."},{"key":"BF02017491_CR82","unstructured":"[1985a] A typed \u03bb-calculus and Girard's model of ptykes.Foundations of logic and linguistics: problems and their solutions (Eds: P. Weingartner and G. Dorn), pp. 245\u2013279, Plenum."},{"key":"BF02017491_CR83","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0168-0072(84)90034-4","volume":"27","author":"J. Pearce","year":"1984","unstructured":"J. Pearce [1984] A constructive consistency proof of a fragment of set theory. Ann. Pure Appl. Logic27 (1984) 25\u201362.","journal-title":"Ann. Pure Appl. Logic"},{"key":"BF02017491_CR84","volume-title":"Ausgezeichnete Folgen f\u00fcr gewisse Abschnitte der zweiten und weiterer Zahlklassen","author":"H. Pfeiffer","year":"1964","unstructured":"H. Pfeiffer [1964] Ausgezeichnete Folgen f\u00fcr gewisse Abschnitte der zweiten und weiterer Zahlklassen. Dissertation, Hannover."},{"key":"BF02017491_CR85","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1007\/BF01982045","volume":"12","author":"H. Pfeiffer","year":"1969","unstructured":"[1969] Ein Bezeichnungssystem f\u00fcr Ordinalzahlen. Arch. Math. Logik Grundlag.12 (1969) 12\u201317.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR86","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1007\/BF01967653","volume":"13","author":"H. Pfeiffer","year":"1970","unstructured":"[1970] Ein Bezeichnungssystem f\u00fcr Ordinalzahlen. Arch. Math. Logik Grundlag.13 (1970) 74\u201390.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR87","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/BF02019775","volume":"15","author":"H. Pfeiffer","year":"1972","unstructured":"[1972] Vergleich zweier Bezeichnungssysteme f\u00fcr Ordinalzahlen. Arch. Math. Logik Grundlag.15 (1972) 41\u201356.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR88","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/BF02025116","volume":"16","author":"H. Pfeiffer","year":"1974","unstructured":"[1974] \u00dcber zwei Bezeichnungssysteme f\u00fcr Ordinalzahlen. Arch. Math. Logik Grundlag.16 (1974) 23\u201336.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR89","unstructured":"W. Pohlers [198?] Ordinal notations based on a hierarchy of inaccessible cardinals. To appear in Ann. Pure Appl. Logic."},{"key":"BF02017491_CR90","unstructured":"D. Schmidt [1972] Topics in mathematical logic. Characterisations of small constructive ordinals; constructive finite number classes. D. Phil. thesis, Oxford."},{"key":"BF02017491_CR91","doi-asserted-by":"crossref","first-page":"305","DOI":"10.2307\/2272156","volume":"40","author":"D. Schmidt","year":"1975","unstructured":"[1975] Bounds for the closure ordinals of replete monotonic increasing functions. J. Symbolic Logic40 (1975) 305\u2013316.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR92","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/BF02007256","volume":"18","author":"D. Schmidt","year":"1976","unstructured":"[1976] Built-up systems of fundamental sequences and hierarchies of number theoretic functions. Arch. Math. Logik Grundlag.18 (1976) 47\u201353; ibid.18 (1976) 145\u2013146.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR93","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/BF01361109","volume":"127","author":"K. Sch\u00fctte","year":"1954","unstructured":"K. Sch\u00fctte [1954] Kennzeichnung von Ordnungszahlen durch rekursiv erkl\u00e4rte Funktionen. Math. Ann.127 (1954) 15\u201332.","journal-title":"Math. Ann."},{"key":"BF02017491_CR94","volume-title":"Beweistheorie","author":"K. Sch\u00fctte","year":"1960","unstructured":"[1960]Beweistheorie. Springer, Berlin. (Completely revised English edition 1977)"},{"key":"BF02017491_CR95","unstructured":"[1963] Lecture Notes in Mathematical Logic. Vol. 2. Penn. State University."},{"key":"BF02017491_CR96","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1016\/S0049-237X(08)71694-X","volume-title":"Formal systems and recursive functions","author":"K. Sch\u00fctte","year":"1965","unstructured":"[1965] Predicative well-orderings.Formal systems and recursive functions (Eds: J.N. Crossley and M.A.E. Dummett), pp. 280\u2013303, North-Holland, Amsterdam."},{"key":"BF02017491_CR97","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1007\/BF01967820","volume":"11","author":"K. Sch\u00fctte","year":"1968","unstructured":"[1968\u201369] Ein konstruktives System von Ordnungszahlen. Arch. Math. Logik Grundlag.11 (1968) 126\u2013137; ibid.12 (1969) 3\u201311.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR98","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF02276805","volume":"17","author":"K. Sch\u00fctte","year":"1976","unstructured":"[1976] Einf\u00fchrung der Normalfunktionen\u0398 \u03b1 ohne Auswahlaxiom und ohne Regularit\u00e4tsbedingung. Arch. Math. Logik Grundlag.17 (1976) 171\u2013178.","journal-title":"Arch. Math. Logik Grundlag."},{"key":"BF02017491_CR99","volume-title":"Proceedings of the Herbrand Symposium, Logic Colloquium '81","year":"1982","unstructured":"J. Stern (Ed.) [1982]Proceedings of the Herbrand Symposium, Logic Colloquium '81. North-Holland, Amsterdam."},{"key":"BF02017491_CR100","volume-title":"The collected papers of Gerhard Gentzen","author":"M. E. Szabo","year":"1969","unstructured":"M. E. Szabo [1969]The collected papers of Gerhard Gentzen. North-Holland, Amsterdam."},{"key":"BF02017491_CR101","doi-asserted-by":"crossref","first-page":"386","DOI":"10.2969\/jmsj\/00940386","volume":"9","author":"G. Takeuti","year":"1957","unstructured":"G. Takeuti [1957] Ordinal diagrams. J. Math. Soc. Japan9 (1957) 386\u2013394.","journal-title":"J. Math. Soc. Japan"},{"key":"BF02017491_CR102","volume-title":"Proof theory","author":"G. Takeuti","year":"1975","unstructured":"[1975]Proof theory. North-Holland, Amsterdam."},{"key":"BF02017491_CR103","doi-asserted-by":"crossref","unstructured":"J. van de Wiele [1982] Recursive dilators and generalized recursions. In J. Stern [1982], pp. 325\u2013332.","DOI":"10.1016\/S0049-237X(08)71893-7"},{"key":"BF02017491_CR104","doi-asserted-by":"crossref","unstructured":"J. Vauzeilles [1982] Functors and ordinal notations III: Dilators and gardens. In J. Stern [1982], pp. 333\u2013364.","DOI":"10.1016\/S0049-237X(08)71894-9"},{"key":"BF02017491_CR105","doi-asserted-by":"crossref","first-page":"331","DOI":"10.2307\/2274218","volume":"50","author":"J. Vauzeilles","year":"1985","unstructured":"[1985] Functors and ordinal notations IV: The Howard ordinal and the functor\u039b. J. Symbolic Logic50 (1985) 331\u2013338.","journal-title":"J. Symbolic Logic"},{"key":"BF02017491_CR106","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1090\/S0002-9947-1908-1500814-9","volume":"9","author":"O. Veblen","year":"1908","unstructured":"O. Veblen [1908] Continuous increasing functions of finite and transfinite ordinals. Trans. Amer. Math. Soc.9 (1908) 280\u2013292.","journal-title":"Trans. Amer. Math. Soc."},{"key":"BF02017491_CR107","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1090\/pspum\/042\/791073","volume":"42","author":"S. S. Wainer","year":"1985","unstructured":"S. S. Wainer [1985] The \u201cslow-growing\u201d\u03a01\/2 approach to hierarchies.Proc. Symposia in Pure Math. 42, pp. 487\u2013502, Amer. Math. Soc., Providence, RI.","journal-title":"Proc. Symposia in Pure Math."},{"key":"BF02017491_CR108","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/BFb0076232","volume-title":"Recursion Theory Week","author":"S. S. Wainer","year":"1985","unstructured":"[1985a] Subrecursive ordinals.Recursion Theory Week, pp. 405\u2013418, Lecture Notes in Mathematics, 1141, Springer, Berlin."},{"key":"BF02017491_CR109","volume-title":"Relations between some-hierarchies of ordinal functions and functionals","author":"R. W. Weyrauch","year":"1972","unstructured":"R. W. Weyrauch [1972] Relations between some-hierarchies of ordinal functions and functionals. Ph. D. thesis, Stanford Univ., California."}],"container-title":["Archiv f\u00fcr Mathematische Logik und Grundlagenforschung"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02017491.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF02017491\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02017491","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,15]],"date-time":"2021-07-15T11:34:57Z","timestamp":1626348897000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF02017491"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,12]]},"references-count":109,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1987,12]]}},"alternative-id":["BF02017491"],"URL":"https:\/\/doi.org\/10.1007\/bf02017491","relation":{},"ISSN":["0933-5846","1432-0665"],"issn-type":[{"value":"0933-5846","type":"print"},{"value":"1432-0665","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,12]]}}}