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Systems of predicative analysis1

  • Solomon Feferman (a1)
Extract

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.

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1

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

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[1]Feferman, S., Classifications of recursive functions by means of hierarchies, Transactions of the American Mathematical Society, vol. 104 (1962), pp. 101122.
[2]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259316.
[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.
[4]Gödel, K., Russell's mathematical logic, The philosophy of Bertrand Russell, New York, 1944, pp. 125153.
[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).
[6]Grzegorczyck, A., Elementary definable analysis, Fundamenta mathematica, vol. 41 (1955), pp. 311338.
[7]Kleene, S. C., Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 193213.
[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.
[9]Kleene, S. C., Quantification of number-theoretic functions, Compositio Mathematica, vol. 14 (1959), pp. 2340.
[10]Kreisel, G., La predicativité, Bulletin de la Société Mathématique de France, vol. 88 (1960), pp. 371391.
[11]Kreisel, G., Ordinal logics and the characterization of informal concepts of proof, Proceedings of the International Congress of Mathematicians (1958); pp. 289299.
[12]Kreisel, G., Ordinals of ramified analysis (abstract), this Journal, vol. 25 (1960), pp. 390391.
[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.
[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.
[15]Lorenzen, P., Logical reflection and formalism, this Journal, vol. 23 (1958), pp. 241249.
[16]Lorenzen, P. and Myhill, J., Constructive definition of certain analytic sets of numbers, this Journal, vol. 24 (1959), pp. 3749.
[17]Schütte, K., Beweistheorie, Springer-Verlag, Berlin, 1960.
[18]Schütte, K., Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik, to appear.
[19]Schütte, K., Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen, Mathematische Annalen, vol. 127 (1954), pp. 1532.
[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.
[21]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151163.
[22]Tait, W. W., The ε-substitution method, to appear.
[23]Takeuti, G., On the fundamental conjecture of GLC. VI, Proceedings of the Japan Academy, vol. 37 (1961), pp. 440443.
[24]Takeuti, G., On the inductive definition with quantifiers of second order, Journal of the Mathematical Society of Japan, vol. 13 (1961), pp. 333341.
[25]Veblen, O., Continuous increasing functions of finite and transfinite ordinals, Transactions of the American Mathematical Society, vol. 9 (1908), pp. 280292.
[26]Wang, H., A survey of mathematical logic, Amsterdam, North-Holland Publishing Company, 1963.
[27]Wang, H., Ordinal numbers and predicative set theory, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 216239.
[28]Wang, H., The formalization of mathematics, this Journal, vol. 19 (1954), pp. 241266.
[29]Weyl, H., Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis, Leipzig (1918), iv + 84 pp.
[30]Whitehead, A. N. and Russell, B., Principia mathematica, vol. I, Cambridge University Press, Cambridge, 2nd edition, 1925.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
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