{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T20:04:52Z","timestamp":1781381092378,"version":"3.54.1"},"reference-count":198,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2017,2,20]],"date-time":"2017-02-20T00:00:00Z","timestamp":1487548800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2017,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\u00e9 to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak K\u00f6nig\u2019s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.<\/jats:p>","DOI":"10.1017\/s1755020316000411","type":"journal-article","created":{"date-parts":[[2017,2,20]],"date-time":"2017-02-20T07:50:56Z","timestamp":1487577056000},"page":"357-396","source":"Crossref","is-referenced-by-count":12,"title":["THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC"],"prefix":"10.1017","volume":"10","author":[{"given":"WALTER","family":"DEAN","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"SEAN","family":"WALSH","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2017,2,20]]},"reference":[{"key":"S1755020316000411_ref176","volume-title":"Constructivism in Mathematics, An Introduction, Vol. 2","author":"Troelstra","year":"1988"},{"key":"S1755020316000411_ref172","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71195-9"},{"key":"S1755020316000411_ref194","volume-title":"Principia Mathematica","author":"Whitehead","year":"1910"},{"key":"S1755020316000411_ref177","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70145-4"},{"key":"S1755020316000411_ref195","author":"Whitehead","year":"1962"},{"key":"S1755020316000411_ref185","doi-asserted-by":"publisher","DOI":"10.1007\/BF01343173"},{"key":"S1755020316000411_ref174","first-page":"261","article-title":"Der Wahrheitsbegriff in den formalisierten Sprache","volume":"1","author":"Tarski","year":"1936","journal-title":"Studia Philosophica"},{"key":"S1755020316000411_ref190","doi-asserted-by":"crossref","DOI":"10.1515\/9783112451144","volume-title":"Das Kontinuum. Kritische Untersuchungen \u00fcber die Grundlagen der Analysis","author":"Weyl","year":"1918"},{"key":"S1755020316000411_ref169","doi-asserted-by":"publisher","DOI":"10.2307\/2272271"},{"key":"S1755020316000411_ref192","first-page":"1","article-title":"Die heutige Erkenntnislage in der Mathematik","volume":"1","author":"Weyl","year":"1926","journal-title":"Symposion; philosophische Zeitschrift f\u00fcr Forschung und Ausspr\u00e4che"},{"key":"S1755020316000411_ref184","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-015-9375-5"},{"key":"S1755020316000411_ref186","doi-asserted-by":"publisher","DOI":"10.2307\/2267732"},{"key":"S1755020316000411_ref171","doi-asserted-by":"publisher","DOI":"10.1016\/S1874-5857(04)80024-4"},{"key":"S1755020316000411_ref187","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70304-0"},{"key":"S1755020316000411_ref183","doi-asserted-by":"publisher","DOI":"10.1017\/S1755020313000397"},{"key":"S1755020316000411_ref157","doi-asserted-by":"publisher","DOI":"10.2307\/2964680"},{"key":"S1755020316000411_ref144","doi-asserted-by":"publisher","DOI":"10.2307\/3602860"},{"key":"S1755020316000411_ref188","volume-title":"From Mathematics to Philosophy","author":"Wang","year":"1974"},{"key":"S1755020316000411_ref166","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"S1755020316000411_ref150","first-page":"263","article-title":"La th\u00e9orie des types logiques","volume":"18","author":"Russell","year":"1910","journal-title":"Revue de M\u00e9taphysique et de Morale"},{"key":"S1755020316000411_ref162","doi-asserted-by":"publisher","DOI":"10.1007\/s11229-004-6300-9"},{"key":"S1755020316000411_ref148","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-4.1.29"},{"key":"S1755020316000411_ref149","doi-asserted-by":"publisher","DOI":"10.2307\/2369948"},{"key":"S1755020316000411_ref164","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(09)70158-2"},{"key":"S1755020316000411_ref155","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66473-1"},{"key":"S1755020316000411_ref14","first-page":"357","article-title":"Semantic construction of intuitionistic logic","volume":"19","author":"Beth","year":"1956","journal-title":"Mededelingen der Koninklijke Nederandse Akademie van Wetenschappen, Afd. Letterkunde"},{"key":"S1755020316000411_ref191","doi-asserted-by":"publisher","DOI":"10.1007\/BF02102305"},{"key":"S1755020316000411_ref42","doi-asserted-by":"publisher","DOI":"10.2307\/2964544"},{"key":"S1755020316000411_ref28","volume-title":"Was sind und was sollen die Zahlen?","author":"Dedekind","year":"1888"},{"key":"S1755020316000411_ref181","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-014-0384-9"},{"key":"S1755020316000411_ref29","doi-asserted-by":"publisher","DOI":"10.1093\/0195148770.003.0005"},{"key":"S1755020316000411_ref132","first-page":"663","article-title":"On models of axiomatic set theory","volume":"4","author":"Mostowski","year":"1956","journal-title":"Bulletin de l\u2019Acad\u00e9mie Polonaise des Sciences"},{"key":"S1755020316000411_ref39","doi-asserted-by":"publisher","DOI":"10.1093\/0195148770.003.0019"},{"key":"S1755020316000411_ref85","doi-asserted-by":"publisher","DOI":"10.1070\/RM1974v029n05ABEH001298"},{"key":"S1755020316000411_ref45","unstructured":"Friedman H . (1967). Subsystems of set theory and analysis. Dissertation, MIT, Unpublished."},{"key":"S1755020316000411_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2003.10.009"},{"key":"S1755020316000411_ref3","unstructured":"Addison J. W . (1954). On Some Points of the Theory of Recursive Functions. Dissertation, University of Wisconsin at Madison."},{"key":"S1755020316000411_ref175","volume-title":"Logic, Semantics, and Metamathematics","author":"Tarski","year":"1956"},{"key":"S1755020316000411_ref40","doi-asserted-by":"publisher","DOI":"10.2307\/420965"},{"key":"S1755020316000411_ref182","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1515\/crll.1925.154.219","article-title":"Eine Axiomatisierung der Mengenlehre","volume":"154","author":"Von Neumann","year":"1925","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"S1755020316000411_ref67","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-7066-3_5"},{"key":"S1755020316000411_ref173","doi-asserted-by":"publisher","DOI":"10.2307\/2026089"},{"key":"S1755020316000411_ref133","first-page":"178","volume-title":"Constructivity in Mathematics, Studies in Logic and the Foundations of Mathematics","author":"Mostowski","year":"1959"},{"key":"S1755020316000411_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01447860"},{"key":"S1755020316000411_ref168","first-page":"88","article-title":"Sur un d\u00e9finition des ensembles measurables B sans nombres transfinis","volume":"164","author":"Souslin","year":"1917","journal-title":"Comptes rendus hebdomadaires des s\u00e9ances de l\u2019Acad\u00e9mie des Sciences"},{"key":"S1755020316000411_ref4","doi-asserted-by":"crossref","first-page":"123","DOI":"10.4064\/fm-46-2-123-135","article-title":"Separation principles in the hierarchies of classical and effective descriptive set theory","volume":"46","author":"Addison","year":"1959","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref16","volume-title":"Le\u00e7ons sur la Th\u00e8orie des Fonctions","author":"Borel","year":"1898"},{"key":"S1755020316000411_ref130","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/155"},{"key":"S1755020316000411_ref7","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.59.3.708"},{"key":"S1755020316000411_ref26","volume-title":"Introduction to Mathematical Logic","author":"Church","year":"1956"},{"key":"S1755020316000411_ref127","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9478-5"},{"key":"S1755020316000411_ref27","doi-asserted-by":"publisher","DOI":"10.2307\/2272393"},{"key":"S1755020316000411_ref111","doi-asserted-by":"publisher","DOI":"10.2307\/2964110"},{"key":"S1755020316000411_ref53","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/257\/04031"},{"key":"S1755020316000411_ref38","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195080308.001.0001","volume-title":"In the Light of Logic","author":"Feferman","year":"1998"},{"key":"S1755020316000411_ref43","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5049-0"},{"key":"S1755020316000411_ref11","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2307\/2268862","article-title":"A system of axiomatic set theory: Part I","volume":"2","author":"Bernays","year":"1937","journal-title":"Journal of Symbolic Logic"},{"key":"S1755020316000411_ref8","first-page":"177","article-title":"Second order arithmetic and related topics","volume":"6","author":"Apt","year":"1973","journal-title":"Annals of Pure and Applied Logic"},{"key":"S1755020316000411_ref165","doi-asserted-by":"publisher","DOI":"10.1017\/S0022481200028309"},{"key":"S1755020316000411_ref54","first-page":"325","article-title":"Higher set theory and mathematical practice","volume":"2","author":"Friedman","year":"1970\/1971","journal-title":"Annals of Pure and Applied Logic"},{"key":"S1755020316000411_ref178","doi-asserted-by":"publisher","DOI":"10.1017\/CCOL0521631785.009"},{"key":"S1755020316000411_ref47","first-page":"235","volume-title":"Proceedings of the International Congress of Mathematicians, 1974","volume":"1","author":"Friedman","year":"1975a"},{"key":"S1755020316000411_ref55","first-page":"577","article-title":"Set existence","volume":"8","author":"Gandy","year":"1960","journal-title":"Bulletin de l\u2019Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S1755020316000411_ref51","unstructured":"Friedman H . (2007). Interpretations, According to Tarski. 19th Annual Tarski Lectures. Available at: http:\/\/u.osu.edu\/friedman.8\/foundational-adventures\/downloadable-manuscripts\/."},{"key":"S1755020316000411_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511750762.009"},{"key":"S1755020316000411_ref46","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066789"},{"key":"S1755020316000411_ref58","volume-title":"Collected Works. Volume I. Publications 1929\u20131936","author":"G\u00f6del","year":"1986"},{"key":"S1755020316000411_ref71","first-page":"69","volume-title":"Constructivity in Mathematics","author":"Heyting","year":"1959"},{"key":"S1755020316000411_ref131","doi-asserted-by":"crossref","first-page":"110","DOI":"10.4064\/fm-37-1-111-124","article-title":"Some impredicative definitions in the axiomatic set-theory","volume":"37","author":"Mostowski","year":"1950","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01594179"},{"key":"S1755020316000411_ref136","first-page":"406","volume-title":"Proof Theory, Second Edition","volume":"81","author":"Pohlers","year":"1987"},{"key":"S1755020316000411_ref134","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70771-7"},{"key":"S1755020316000411_ref179","first-page":"1","volume-title":"Cambridge Summer School in Mathematical Logic","author":"van Dalen","year":"1973"},{"key":"S1755020316000411_ref63","first-page":"241","article-title":"La th\u00e9orie des ensembles","volume":"16","author":"Hadamard","year":"1905b","journal-title":"Revue g\u00e9n\u00e9rale des sciences pures et appliqu\u00e9es"},{"key":"S1755020316000411_ref12","doi-asserted-by":"publisher","DOI":"10.2307\/2266303"},{"key":"S1755020316000411_ref17","first-page":"15","volume-title":"Atti del IV Congresso Internazionale dei Matematici (Roma, 6\u201311 Aprile 1908)","volume":"1","author":"Borel","year":"1909"},{"key":"S1755020316000411_ref87","first-page":"679","volume-title":"Proceedings of the International Congress of Mathematicians, 1950","author":"Kleene","year":"1952b"},{"key":"S1755020316000411_ref37","first-page":"147","volume-title":"Philosophy of Mathematics","volume":"20","author":"Feferman","year":"1993"},{"key":"S1755020316000411_ref145","doi-asserted-by":"publisher","DOI":"10.2307\/2266971"},{"key":"S1755020316000411_ref147","first-page":"627","article-title":"Les paradoxes de la logique","volume":"14","author":"Russell","year":"1906","journal-title":"Revue de m\u00e9taphysique et de morale"},{"key":"S1755020316000411_ref49","first-page":"551","article-title":"Systems of second order arithmetic with restricted induction. I and II (abstracts)","volume":"41","author":"Friedman","year":"1976","journal-title":"The Journal of Symbolic Logic"},{"key":"S1755020316000411_ref193","volume-title":"Gesammelte Abhandlungen.","author":"Weyl","year":"1968"},{"key":"S1755020316000411_ref106","first-page":"621","article-title":"Analysis of the Cantor-Bendixson theorem by means of the analytic hierarchy","volume":"7","author":"Kreisel","year":"1959a","journal-title":"Bulletin de l\u2019Acad\u00e9mie Polonaise des Sciences. S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S1755020316000411_ref10","first-page":"326","article-title":"Die Philosophie der Mathematik und die Hilbertsche Beweistheorie","volume":"4","author":"Bernays","year":"1930","journal-title":"Bl\u00e4tter f\u00fcr deutsche Philosophie"},{"key":"S1755020316000411_ref116","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/028.1\/0434781"},{"key":"S1755020316000411_ref156","first-page":"117","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"5","author":"Scott","year":"1962"},{"key":"S1755020316000411_ref78","volume-title":"Grundlagen der Mathematik","volume":"II.","author":"Hilbert","year":"1939"},{"key":"S1755020316000411_ref52","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(83)90012-X"},{"key":"S1755020316000411_ref112","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71525-8"},{"key":"S1755020316000411_ref41","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0091895"},{"key":"S1755020316000411_ref31","unstructured":"Dyson V. & Kreisel G . (1961). Analysis of Beth\u2019s semantic construction of intuitionistic logic. Technical Report DA-04-200-ORD-997, Applied Mathematics and Statistical Laboratories, Stanford University."},{"key":"S1755020316000411_ref167","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"},{"key":"S1755020316000411_ref74","volume-title":"Lectures on the Foundations of Arithmetic and Logic: 1917\u20131933","volume":"3","author":"Hilbert","year":"2013"},{"key":"S1755020316000411_ref180","volume-title":"From Frege to G\u00f6del : A Source Book in Mathematical Logic, 1879\u20131931","author":"van Heijenoort","year":"1967"},{"key":"S1755020316000411_ref151","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12013-2"},{"key":"S1755020316000411_ref138","first-page":"294","article-title":"Les math\u00e9matiques et la logique","volume":"14","author":"Poincar\u00e9","year":"1906","journal-title":"Revue de m\u00e9taphysique et de morale"},{"key":"S1755020316000411_ref92","doi-asserted-by":"crossref","first-page":"197","DOI":"10.4099\/jjm1924.15.0_197","article-title":"Sur l\u2019uniformisation des compl\u00e9mentaires analytiques et les ensembles projectifs de la seconde classe","volume":"15","author":"Kond\u00f4","year":"1939","journal-title":"Japanese Journal of Mathematics"},{"key":"S1755020316000411_ref107","unstructured":"Kreisel G . (1959b). Review of Kond\u00f4 (1958). Mathematical Reviews. MR0113806 (22 #4638)."},{"key":"S1755020316000411_ref113","doi-asserted-by":"publisher","DOI":"10.2307\/2270324"},{"key":"S1755020316000411_ref110","doi-asserted-by":"publisher","DOI":"10.1016\/S1385-7258(62)50029-2"},{"key":"S1755020316000411_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01449946"},{"key":"S1755020316000411_ref197","doi-asserted-by":"publisher","DOI":"10.1007\/BF01450054"},{"key":"S1755020316000411_ref13","first-page":"572","article-title":"Semantical considerations on intuitionistic mathematics","volume":"9","author":"Beth","year":"1947","journal-title":"Indagationes Mathematicae"},{"key":"S1755020316000411_ref23","article-title":"Effectively Closed Sets. To appear in the ASL series","author":"Cenzer","year":"2012","journal-title":"Lecture Notes in Logic"},{"key":"S1755020316000411_ref25","volume-title":"Introduction to Mathematical Logic","author":"Church","year":"1944"},{"key":"S1755020316000411_ref153","doi-asserted-by":"publisher","DOI":"10.1007\/BF01972460"},{"key":"S1755020316000411_ref70","volume-title":"Intuitionism. An introduction","author":"Heyting","year":"1956"},{"key":"S1755020316000411_ref120","volume-title":"Mathematical Logic and Hilbert\u2019s \u03b5-Symbol","author":"Leisenring","year":"1969"},{"key":"S1755020316000411_ref103","unstructured":"Kreisel G . (1955). Review of kleene (1955). Mathematical Reviews. MR0070593 (17,4f)."},{"key":"S1755020316000411_ref198","doi-asserted-by":"publisher","DOI":"10.1007\/BF01449999"},{"key":"S1755020316000411_ref20","volume-title":"Collected Works 1. Philosophy and Foundations of Mathematics","author":"Brouwer","year":"1975"},{"key":"S1755020316000411_ref129","volume-title":"Descriptive Set Theory","volume":"100","author":"Moschovakis","year":"1980"},{"key":"S1755020316000411_ref44","doi-asserted-by":"publisher","DOI":"10.1007\/BF00376449"},{"key":"S1755020316000411_ref9","volume-title":"Computable Structures and the Hyperarithmetical Hierarchy","volume":"144","author":"Ash","year":"2000"},{"key":"S1755020316000411_ref93","first-page":"1841","article-title":"Sur la nommabilit\u00e9 d\u2019ensembles","volume":"242","author":"Kond\u00f4","year":"1956","journal-title":"Comptes Rendus Hebdomadaires des s\u00e9ances de l\u2019Acad\u00e9mie des Sciences"},{"key":"S1755020316000411_ref22","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1122038914"},{"key":"S1755020316000411_ref102","doi-asserted-by":"publisher","DOI":"10.1093\/bjps\/IV.14.107"},{"key":"S1755020316000411_ref24","doi-asserted-by":"publisher","DOI":"10.2307\/2266170"},{"key":"S1755020316000411_ref66","doi-asserted-by":"publisher","DOI":"10.2307\/2266326"},{"key":"S1755020316000411_ref33","doi-asserted-by":"publisher","DOI":"10.2307\/2269764"},{"key":"S1755020316000411_ref84","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S1755020316000411_ref69","doi-asserted-by":"publisher","DOI":"10.2307\/2267403"},{"key":"S1755020316000411_ref163","unstructured":"Simpson S. G . (1973). Notes on subsystems of analysis. Unpublished, typewritten, Berkeley, 38 pages."},{"key":"S1755020316000411_ref72","doi-asserted-by":"publisher","DOI":"10.1007\/BF02940589"},{"key":"S1755020316000411_ref73","doi-asserted-by":"publisher","DOI":"10.1007\/BF01206605"},{"key":"S1755020316000411_ref79","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-0224-6"},{"key":"S1755020316000411_ref77","volume-title":"Grundlagen der Mathematik","volume":"I","author":"Hilbert","year":"1934"},{"key":"S1755020316000411_ref56","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7402-1"},{"key":"S1755020316000411_ref86","volume-title":"Introduction to Metamathematics","volume":"1","author":"Kleene","year":"1952a"},{"key":"S1755020316000411_ref62","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.761"},{"key":"S1755020316000411_ref64","volume-title":"Metamathematics of First-Order Arithmetic","author":"H\u00e1jek","year":"1998"},{"key":"S1755020316000411_ref57","first-page":"39","article-title":"La th\u00e9orie des ensembles en France avant la crise de 1905: Baire, Borel, Lebesgue \u2026 et tous les autres","volume":"1","author":"Gispert","year":"1995","journal-title":"Revue d\u2019Histoire des Math\u00e9matiques"},{"key":"S1755020316000411_ref5","first-page":"26","volume-title":"Logic, Methodology and Philosophy of Science (Proceedings of the 1960 International Congress)","author":"Addison","year":"1962"},{"key":"S1755020316000411_ref60","doi-asserted-by":"crossref","first-page":"311","DOI":"10.4064\/fm-41-2-311-338","article-title":"Elementarily definable analysis","volume":"41","author":"Grzegorczyk","year":"1955","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref65","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0244049-7"},{"key":"S1755020316000411_ref76","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-41928-1"},{"key":"S1755020316000411_ref59","volume-title":"Collected Works. Vol. II. Publications 1938\u20131974","author":"G\u00f6del","year":"1990"},{"key":"S1755020316000411_ref36","doi-asserted-by":"crossref","first-page":"364","DOI":"10.2307\/2274509","article-title":"Hilbert\u2019s program relativized: Proof-theoretical and foundational reductions","volume":"53","author":"Feferman","year":"1988","journal-title":"The Journal of Symbolic Logic"},{"key":"S1755020316000411_ref75","first-page":"809","volume-title":"Grundz\u00fcge der theoretischen Logik","author":"Hilbert","year":"1928"},{"key":"S1755020316000411_ref118","doi-asserted-by":"crossref","first-page":"76","DOI":"10.4064\/fm-3-1-76-108","article-title":"Une m\u00e9thode d\u2019\u00e9limination des nombres transfinis des raisonnements math\u00e9matiques","volume":"3","author":"Kuratowski","year":"1922","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref90","doi-asserted-by":"publisher","DOI":"10.1057\/9780230245198_5"},{"key":"S1755020316000411_ref170","unstructured":"Steel J . (1977). Determinateness and subsystems of analysis.Dissertation, University of California, Berkeley, Unpublished."},{"key":"S1755020316000411_ref88","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1955-09896-3"},{"key":"S1755020316000411_ref68","doi-asserted-by":"publisher","DOI":"10.2307\/2267044"},{"key":"S1755020316000411_ref94","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4099\/jjm1924.28.0_1","article-title":"Sur les ensembles nommables et le fondement de l\u2019analyse math\u00e9matique. I","volume":"28","author":"Kond\u00f4","year":"1958","journal-title":"Japanese Journal of Mathematics"},{"key":"S1755020316000411_ref105","doi-asserted-by":"publisher","DOI":"10.2307\/2964396"},{"key":"S1755020316000411_ref83","doi-asserted-by":"publisher","DOI":"10.1070\/RM2003v058n05ABEH000666"},{"key":"S1755020316000411_ref97","first-page":"121","article-title":"\u00dcber eine Schlussweise aus dem Endlichen ins Unendliche","volume":"3","author":"K\u00f6nig","year":"1927","journal-title":"Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae"},{"key":"S1755020316000411_ref98","volume-title":"Theorie der endlichen und unendlichen Graphen","author":"K\u00f6nig","year":"1936"},{"key":"S1755020316000411_ref80","first-page":"243","article-title":"Der angebliche circulus vitiosus und die sogenannte Grundlagenkrise in der Analysis","volume":"78","author":"H\u00f6lder","year":"1926","journal-title":"Berichte \u00fcber die Verhandlungen der S\u00e4chsischen Akademie der Wissenschaft zu Leipzig, Philologisch-Historische Klasse"},{"key":"S1755020316000411_ref81","volume-title":"Set Theory","author":"Jech","year":"2003"},{"key":"S1755020316000411_ref101","doi-asserted-by":"crossref","first-page":"265","DOI":"10.4064\/fm-37-1-265-285","article-title":"Note on arithmetic models for consistent formulae of the predicate calculus","volume":"37","author":"Kreisel","year":"1950","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref48","unstructured":"Friedman H . (1975b). The Analysis of Mathematical Texts, and their Calibration in Terms of Intrinsic Strength; and the Logical Strength of Mathematical Statements. Unpublished manuscripts. https:\/\/u.osu.edu\/friedman.8\/foundational-adventures\/downloadable-manuscripts\/."},{"key":"S1755020316000411_ref95","first-page":"61","article-title":"Le fondement constructif du calcul infinit\u00e9simal","volume":"12","author":"Kond\u00f4","year":"1960","journal-title":"Osaka Journal of Mathematics"},{"key":"S1755020316000411_ref108","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1554"},{"key":"S1755020316000411_ref89","first-page":"23","article-title":"Quantification of number-theoretic functions","volume":"14","author":"Kleene","year":"1959","journal-title":"Compositio Mathematica"},{"key":"S1755020316000411_ref104","doi-asserted-by":"publisher","DOI":"10.2307\/2964291"},{"key":"S1755020316000411_ref91","volume-title":"Applied Proof Theory: Proof Interpretations and Their Use in Mathematics","author":"Kohlenbach","year":"2008"},{"key":"S1755020316000411_ref189","first-page":"93","article-title":"\u00dcber die Definitionen der mathematischen Grundbegriffe","volume":"7","author":"Weyl","year":"1910","journal-title":"Mathematisch-naturwissenschaftliche Bl\u00e4tter"},{"key":"S1755020316000411_ref100","first-page":"1","article-title":"Andrzej Grzegorczyk: Logic and philosophy","volume":"81","author":"Krajewski","year":"2007","journal-title":"Topics in Logic, Philosophy and Foundations of Mathematics, and Computer Science: In Recognition of Professor Andrzej Grzegorczyk"},{"key":"S1755020316000411_ref82","first-page":"33","article-title":"${\\rm{\\Pi }}_1^0 $ classes and degrees of theories","volume":"173","author":"Jockusch","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"S1755020316000411_ref30","first-page":"11","volume-title":"Studies in Logic and the Foundations of Mathematics","volume":"129","author":"Drake","year":"1989"},{"key":"S1755020316000411_ref109","first-page":"289","volume-title":"Proceedings of the International Congress of Mathematicians, 1958","author":"Kreisel","year":"1960b"},{"key":"S1755020316000411_ref114","first-page":"121","volume-title":"Intuitionism and Proof Theory","author":"Kreisel","year":"1970a"},{"key":"S1755020316000411_ref122","doi-asserted-by":"crossref","first-page":"48","DOI":"10.4064\/fm-16-1-48-76","article-title":"Analogies entre les ensembles mesurables B et les ensembles analytiques","volume":"1","author":"Lusin","year":"1930a","journal-title":"Fundamenta Mathematicae"},{"key":"S1755020316000411_ref34","first-page":"1","volume-title":"Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies","volume":"897","author":"Feferman","year":"1981"},{"key":"S1755020316000411_ref50","doi-asserted-by":"publisher","DOI":"10.2307\/1971023"},{"key":"S1755020316000411_ref125","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7643-8653-5_10"},{"key":"S1755020316000411_ref35","first-page":"447","volume-title":"Studies in Logic and the Foundations of Mathematics","volume":"81","author":"Feferman","year":"1987"},{"key":"S1755020316000411_ref18","volume-title":"Le\u00e7ons sur la Th\u00e8orie des Fonctions","author":"Borel","year":"1914"},{"key":"S1755020316000411_ref128","doi-asserted-by":"publisher","DOI":"10.1016\/0315-0860(81)90070-7"},{"key":"S1755020316000411_ref121","first-page":"279","article-title":"Sur le probleme de M. \u00c9mile Borel et la m\u00e9thode des r\u00e9solvantes","volume":"181","author":"Lusin","year":"1925","journal-title":"Comptes rendus hebdomadaires des s\u00e9ances de l\u2019Acad\u00e9mie des Sciences"},{"key":"S1755020316000411_ref137","first-page":"815","article-title":"Les math\u00e9matiques et la logique","volume":"13","author":"Poincar\u00e9","year":"1905","journal-title":"Revue de m\u00e9taphysique et de morale"},{"key":"S1755020316000411_ref61","first-page":"43","volume-title":"Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957. Studies in Logic and the Foundations of Mathematics","author":"Grzegorczyk","year":"1959"},{"key":"S1755020316000411_ref117","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0064871"},{"key":"S1755020316000411_ref123","volume-title":"Le\u00e7ons sur les ensembles analytiques et leurs applications","author":"Lusin","year":"1930b"},{"key":"S1755020316000411_ref135","first-page":"372","volume-title":"Reflections on the Foundations of Mathematics (Stanford, CA, 1998)","volume":"15","author":"Parsons","year":"2002"},{"key":"S1755020316000411_ref119","first-page":"139","article-title":"Sur les fonctions repr\u00e9sentables analytiquement","volume":"1","author":"Lebesgue","year":"1905","journal-title":"Journal de Math\u00e9matiques Pures et Appliqu\u00e9es 6e s\u00e9rie"},{"key":"S1755020316000411_ref126","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/pdr022"},{"key":"S1755020316000411_ref96","volume-title":"Selected Works of Motokiti Kond\u00f4","author":"Kond\u00f4","year":"1985"},{"key":"S1755020316000411_ref159","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1286284559"},{"key":"S1755020316000411_ref160","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90030-2"},{"key":"S1755020316000411_ref158","volume-title":"Mathematical Logic","volume":"21","author":"Shoenfield","year":"1967"},{"key":"S1755020316000411_ref142","first-page":"641","article-title":"Degrees of unsolvability \u2013 preliminary report","volume":"54","author":"Post","year":"1948","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S1755020316000411_ref140","doi-asserted-by":"publisher","DOI":"10.1007\/BF02403216"},{"key":"S1755020316000411_ref99","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-8971-2"},{"key":"S1755020316000411_ref141","first-page":"44","volume-title":"Sechs Vortr\u00e4ge \u00fcber ausgew\u00e4hlte Gegenst\u00e4nde aus der reinen Mathematik und mathematischen Physik","author":"Poincar\u00e9","year":"1910"},{"key":"S1755020316000411_ref161","doi-asserted-by":"publisher","DOI":"10.1016\/S1874-5857(09)70012-3"},{"key":"S1755020316000411_ref115","first-page":"489","volume-title":"Intuitionism and Proof Theory","author":"Kreisel","year":"1970b"},{"key":"S1755020316000411_ref139","first-page":"461","article-title":"La logique de l\u2019infini","volume":"17","author":"Poincar\u00e9","year":"1909a","journal-title":"Revue de m\u00e9taphysique et de morale"},{"key":"S1755020316000411_ref146","volume-title":"Equivalents of the Axiom of Choice","author":"Rubin","year":"1963"},{"key":"S1755020316000411_ref152","volume-title":"Beweistheorie","volume":"103","author":"Sch\u00fctte","year":"1960"},{"key":"S1755020316000411_ref196","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026247421383"},{"key":"S1755020316000411_ref143","first-page":"338","article-title":"The foundations of mathematics","volume":"25","author":"Ramsey","year":"1925","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S1755020316000411_ref15","volume-title":"Foundations of Constructive Analysis","volume":"60","author":"Bishop","year":"1967"},{"key":"S1755020316000411_ref32","doi-asserted-by":"publisher","DOI":"10.1080\/0144534031000124134"},{"key":"S1755020316000411_ref124","doi-asserted-by":"publisher","DOI":"10.1023\/A:1026278800910"},{"key":"S1755020316000411_ref154","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71694-X"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020316000411","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,22]],"date-time":"2024-06-22T15:27:33Z","timestamp":1719070053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020316000411\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,20]]},"references-count":198,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,6]]}},"alternative-id":["S1755020316000411"],"URL":"https:\/\/doi.org\/10.1017\/s1755020316000411","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,2,20]]}}}