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Systems of predicative analysis1

Published online by Cambridge University Press:  12 March 2014

Solomon Feferman*
Stanford University


This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.

Part I

§1. Conceptions of sets. Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

Research Article
Copyright © Association for Symbolic Logic 1964

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Text of an invited address delivered to a meeting of the Association for Symbolic Logic at Berkeley, California, on January 26, 1963.


[1]Feferman, S., Classifications of recursive functions by means of hierarchies, Transactions of the American Mathematical Society, vol. 104 (1962), pp. 101122.CrossRefGoogle Scholar
[2]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259316.Google Scholar
[3]Gandy, R., Kreisel, G., and Tait, W., Set existence, Bulletin de l'Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques, vol. 8 (1960), pp. 577582.Google Scholar
[4]Gödel, K., Russell's mathematical logic, The philosophy of Bertrand Russell, New York, 1944, pp. 125153.Google Scholar
[5]Gödel, K., The consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies No. 3, Princeton (1940).Google Scholar
[6]Grzegorczyck, A., Elementary definable analysis, Fundamenta mathematica, vol. 41 (1955), pp. 311338.CrossRefGoogle Scholar
[7]Kleene, S. C., Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193213.CrossRefGoogle Scholar
[8]Kleene, S. C., On the forms of predicates in the theory of constructive ordinals, II, American journal of mathematics, vol. 77 (1955), pp. 405428.CrossRefGoogle Scholar
[9]Kleene, S. C., Quantification of number-theoretic functions, Compositio Mathematica, vol. 14 (1959), pp. 2340.Google Scholar
[10]Kreisel, G., La predicativité, Bulletin de la Société Mathématique de France, vol. 88 (1960), pp. 371391.CrossRefGoogle Scholar
[11]Kreisel, G., Ordinal logics and the characterization of informal concepts of proof, Proceedings of the International Congress of Mathematicians (1958); pp. 289299.Google Scholar
[12]Kreisel, G., Ordinals of ramified analysis (abstract), this Journal, vol. 25 (1960), pp. 390391.Google Scholar
[13]Kreisel, G., The axiom of choice and the class of hyperarithmetic functions, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, ser. A, vol. 65 (also Indagationes mathematicae, vol. 24), (1962), pp. 307319.Google Scholar
[14]Kond, M.ô, Sur les ensembles nommables et le fondement de l'analyse mathématique I, Japanese journal of mathematics, vol. 28 (1958), pp. 1116.CrossRefGoogle Scholar
[15]Lorenzen, P., Logical reflection and formalism, this Journal, vol. 23 (1958), pp. 241249.Google Scholar
[16]Lorenzen, P. and Myhill, J., Constructive definition of certain analytic sets of numbers, this Journal, vol. 24 (1959), pp. 3749.Google Scholar
[17]Schütte, K., Beweistheorie, Springer-Verlag, Berlin, 1960.Google Scholar
[18]Schütte, K., Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik, to appear.Google Scholar
[19]Schütte, K., Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen, Mathematische Annalen, vol. 127 (1954), pp. 1532.CrossRefGoogle Scholar
[20]Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Recursive function theory. Proceedings of symposia in pure mathematics (Providence), vol. 5 (1962), pp. 127.CrossRefGoogle Scholar
[21]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151163.Google Scholar
[22]Tait, W. W., The ε-substitution method, to appear.Google Scholar
[23]Takeuti, G., On the fundamental conjecture of GLC. VI, Proceedings of the Japan Academy, vol. 37 (1961), pp. 440443.CrossRefGoogle Scholar
[24]Takeuti, G., On the inductive definition with quantifiers of second order, Journal of the Mathematical Society of Japan, vol. 13 (1961), pp. 333341.CrossRefGoogle Scholar
[25]Veblen, O., Continuous increasing functions of finite and transfinite ordinals, Transactions of the American Mathematical Society, vol. 9 (1908), pp. 280292.CrossRefGoogle Scholar
[26]Wang, H., A survey of mathematical logic, Amsterdam, North-Holland Publishing Company, 1963.Google Scholar
[27]Wang, H., Ordinal numbers and predicative set theory, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 216239.CrossRefGoogle Scholar
[28]Wang, H., The formalization of mathematics, this Journal, vol. 19 (1954), pp. 241266.Google Scholar
[29]Weyl, H., Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig (1918), iv + 84 pp.Google Scholar
[30]Whitehead, A. N. and Russell, B., Principia mathematica, vol. I, Cambridge University Press, Cambridge, 2nd edition, 1925.Google Scholar
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