Skip to main content
×
Home

THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

  • WALTER DEAN (a1) and SEAN WALSH (a2)
Abstract
Abstract

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 Poincaré 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 König’s 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.

Copyright
Corresponding author
*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF WARWICK COVENTRY CV4 7AL UK E-mail: W.H.Dean@warwick.ac.uk
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE UNIVERSITY OF CALIFORNIA 5100 SOCIAL SCIENCE PLAZA IRVINE, CA 92697-5100 USA E-mail: swalsh108@gmail.com or walsh108@uci.edu
References
Hide All
Ackermann W. (1925). Begründung des “tertium non datur” mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen, 93(1), 136.
Ackermann W. (1937). Die Widerspruchsfreiheit der allgemeinen Mengenlehre. Mathematische Annalen, 114(1), 305315.
Addison J. W. (1954). On Some Points of the Theory of Recursive Functions. Dissertation, University of Wisconsin at Madison.
Addison J. W. (1959). Separation principles in the hierarchies of classical and effective descriptive set theory. Fundamenta Mathematicae, 46, 123135.
Addison J. W. (1962). The theory of hierarchies. In Nagel E., Suppes P., and Tarski A., editors. Logic, Methodology and Philosophy of Science (Proceedings of the 1960 International Congress). Stanford: Stanford University Press, pp. 2637.
Addison J. W. (2004). Tarski’s theory of definability: Common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. Annals of Pure and Applied Logic, 126(1–3), 7792.
Addison J. W. & Moschovakis Y. N. (1968). Some consequences of the axiom of definable determinateness. Proceedings of the National Academy of Sciences of the United States of America, 59, 708712.
Apt K. R. & Marek W. (1973). Second order arithmetic and related topics. Annals of Pure and Applied Logic, 6, 177229.
Ash C. & Knight J. (2000). Computable Structures and the Hyperarithmetical Hierarchy. Studies in Logic and the Foundations of Mathematics, Vol. 144. Amsterdam: North-Holland.
Bernays P. (1930). Die Philosophie der Mathematik und die Hilbertsche Beweistheorie. Blätter für deutsche Philosophie, 4, 326367.
Bernays P. (1937). A system of axiomatic set theory: Part I. Journal of Symbolic Logic, 2(1), 6577.
Bernays P. (1942). A : Part III. Infinity and enumerability. Analysis. The Journal of Symbolic Logic, 7(2), 6589.
Beth E. (1947). Semantical considerations on intuitionistic mathematics. Indagationes Mathematicae, 9, 572577.
Beth E. (1956). Semantic construction of intuitionistic logic. Mededelingen der Koninklijke Nederandse Akademie van Wetenschappen, Afd. Letterkunde, 19(11), 357388.
Bishop E. (1967). Foundations of Constructive Analysis, Vol. 60. New York: McGraw-Hill.
Borel É. (1898). Leçons sur la Thèorie des Fonctions. Paris: Gauthier-Villars.
Borel É. (1909). Sur les principes de la théorie des ensembles. In Feferman S., Parsons C., and Simpson S.G., editors. Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908), Vol. 1. Roma: Tipografia della R. Accademia dei Lincei, pp. 1517.
Borel É. (1914). Leçons sur la Thèorie des Fonctions (second edition). Paris: Gauthier-Villars.
Brouwer L. E. (1927). Über Definitionsbereiche von Funktionen. Mathematische Annalen, 97(1), 6075. Reprinted in Brouwer (1975) and van Heijenoort (1967).
Brouwer L. E. J. (1975). In Heyting A., editor. Collected Works 1. Philosophy and Foundations of Mathematics. Amsterdam: North Holland.
Burgess J. P. (2010). On the outside looking in: A caution about conservativeness. In Kurt Gödel: Essays for His Centennial. Lecture Notes in Logic. Cambridge: Cambridge University Press, pp. 128144.
Caldon P. & Ignjatovic A. (2005). On mathematical instrumentalism. The Journal of Symbolic Logic, 70(3), 778794.
Cenzer D. & Remmel J. B. (2012). Effectively Closed Sets. To appear in the ASL series Lecture Notes in Logic. Unpublished.
Church A. (1940). A formulation of the simple theory of types. Journal of Symbolic Logic, 5(2), 5668.
Church A. (1944). Introduction to Mathematical Logic. Princeton: Princeton University Press.
Church A. (1956). Introduction to Mathematical Logic (second edition). Princeton: Princeton University Press.
Church A. (1976). Comparison of Russell’s resolution of the semantical antinomies with that of Tarski. Journal of Symbolic Logic, 41(4), 747760.
Dedekind R. (1888). Was sind und was sollen die Zahlen? Braunschweig: Vieweg.
Demopoulos W. & Clark P. (2005). The logicism of Frege, Dedekind, and Russell. In Shapiro S., editor. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press, pp. 129165.
Drake F. R. (1989). On the foundations of mathematics in 1987. In Ebbinghaus H.-D., Fernández-Prida J., Garrido M., Lascar D. and Rodríquez Artalejo M., editors. Logic Colloquium ’87. Studies in Logic and the Foundations of Mathematics, Vol. 129. Amsterdam: North-Holland, pp. 1125.
Dyson V. & Kreisel G. (1961). Analysis of Beth’s semantic construction of intuitionistic logic. Technical Report DA-04-200-ORD-997, Applied Mathematics and Statistical Laboratories, Stanford University.
Ebbinghaus H.-D. (2003). Zermelo: Definiteness and the universe of definable sets. History and Philosophy of Logic, 24(3), 197219.
Feferman S. (1964). Systems of predicative analysis. The Journal of Symbolic Logic, 29, 130.
Feferman S. (1981). Preface: How we got from there to here. In Wilfried B., Feferman S., Pohlers W., and Sieg W., editors. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics, Vol. 897. Berlin: Springer, pp. 115.
Feferman S. (1987). Proof theory: A personal report. In Takeuti, G., editor. Proof Theory, Second Edition. Studies in Logic and the Foundations of Mathematics, Vol. 81. Amsterdam: North-Holland, pp. 447481.
Feferman S. (1988). Hilbert’s program relativized: Proof-theoretical and foundational reductions. The Journal of Symbolic Logic, 53(2), 364384.
Feferman S. (1993). What rests on what? The proof-theoretic analysis of mathematics. In Philosophy of Mathematics. Schriftenreihe der Wittgenstein-Gesellschaft, Vol. 20. Vienna: Hölder-Pichler-Tempsky, pp. 147171.
Feferman S. (1998). In the Light of Logic. Oxford: Oxford University Press.
Feferman S. (2005). Predicativity. In Shapiro S., editor. The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford: Oxford University Press, pp. 590624.
Feferman S., Friedman H. M., Maddy P., & Steel J. R. (2000). Does mathematics need new axioms? The Bulletin of Symbolic Logic, 6(4), 401446.
Feferman S. & Sieg W. (1981). Iterated inductive definitions and subsystems of analysis. In Wilfried B., Feferman S., Pohlers W., and Sieg W., editors. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics, Vol. 897. Berlin: Springer, pp. 1677.
Feferman S. & Spector C. (1962). Incompleteness along paths in progressions of theories. The Journal of Symbolic Logic, 27, 383390.
Ferreirós J. (1999). Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Vol. 23. Basel: Birkhäuser.
Franchella M. (1997). On the origins of Dénes König’s infinity lemma. Archive for History of Exact Sciences, 51(1), 327.
Friedman H. (1967). Subsystems of set theory and analysis. Dissertation, MIT, Unpublished.
Friedman H. (1973). Countable models of set theories. In Mathias A. and Rogers H., editors. Cambridge Summer School in Mathematical Logic. Berlin: Springer, pp. 539573.
Friedman H. (1975a). Some systems of second-order arithmetic and their use. In James R. D., editor. Proceedings of the International Congress of Mathematicians, 1974, Vol. 1. Vancouver: Canadian Mathematical Congress, pp. 235242.
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/.
Friedman H. (1976). Systems of second order arithmetic with restricted induction. I and II (abstracts). The Journal of Symbolic Logic, 41(2), 551560.
Friedman H. (1977). Set theoretic foundations for constructive analysis. Annals of Mathematics, 105(1), 128.
Friedman H. (2007). Interpretations, According to Tarski. 19th Annual Tarski Lectures. Available at: http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/.
Friedman H., Simpson S., & Smith R. (1983). Countable algebra and set existence axioms. Annals of Pure and Applied Logic, 25(2), 141181.
Friedman H. & Simpson S. G. (2000). Issues and problems in reverse mathematics. In Cholak P. A., Lempp S., Lerman M., and Shore R. A., editors. Computability Theory and its Applications. Contemporary Mathematics, Vol. 257. Providence: American Mathematical Society, pp. 127144.
Friedman H. M. (1970/1971). Higher set theory and mathematical practice. Annals of Pure and Applied Logic, 2(3), 325357.
Gandy R. O., Kreisel G., & Tait W. W. (1960). Set existence. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 8, 577582.
Garciadiego A. R. D. (1992). Bertrand Russell and the Origins of the Set-Theoretic ‘Paradoxes’. Boston: Birkhäuser.
Gispert H. (1995). La théorie des ensembles en France avant la crise de 1905: Baire, Borel, Lebesgue … et tous les autres. Revue d’Histoire des Mathématiques, 1(1), 3981.
Gödel K. (1986). In Feferman S. et al., editors. Collected Works. Volume I. Publications 1929–1936. New York: Clarendon.
Gödel K. (1990). In Feferman, S. et al., editors. Collected Works. Vol. II. Publications 1938–1974. New York: Clarendon.
Grzegorczyk A. (1955). Elementarily definable analysis. Fundamenta Mathematicae, 41, 311338.
Grzegorczyk A. (1959). Some approaches to constructive analysis. In Heyting A., editor. Constructivity in Mathematics: Proceedings of the Colloquium held at Amsterdam, 1957. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, pp. 4361.
Hadamard J. (1905a). Cinq lettres sur la théorie des ensembles. Bulletin de la Société mathématique de France, 33, 261273.
Hadamard J. (1905b). La théorie des ensembles. Revue générale des sciences pures et appliquées, 16, 241242.
Hájek P. & Pudlák P. (1998). Metamathematics of First-Order Arithmetic. Berlin: Springer.
Harrison J. (1968). Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, 131, 526543.
Hasenjaeger G. (1953). Eine Bemerkung zu Henkin’s Beweis für die vollständigkeit des Prädikatenkalküls der ersten Stufe. The Journal of Symbolic Logic, 18(01), 4248.
Hazen A. P. (1983). Predicative logics. In Gabbay D. and Guenthner F., editors. Handbook of Philosophical Logic. Volume I: Elements of Classical Logic. Dordrecht: Reidel, pp. 331407.
Henkin L. (1949). The completeness of the first-order functional calculus. The Journal of Symbolic Logic, 14(3), 159166.
Henkin L. (1953). Banishing the rule of substitution for functional variables. The Journal of Symbolic Logic, 18, 201208.
Heyting A. (1956). Intuitionism. An introduction. Amsterdam: North-Holland.
Heyting A. (1959). Some remarks on intuitionism. In Heyting A., editor. Constructivity in Mathematics. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, pp. 6971.
Hilbert D. (1922). Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1(1), 157177.
Hilbert D. (1926). Über das Unendliche. Mathematische Annalen, 95, 161190.
Hilbert D. (2013). In Ewald W. and Sieg W., editors. Lectures on the Foundations of Arithmetic and Logic: 1917–1933. David Hilbert’s Foundational Lectures, Vol. 3. Berlin: Springer.
Hilbert D. & Ackermann W. (1928). Grundzüge der theoretischen Logik. Berlin: Springer. Reprinted in Hilbert (2013), pp. 809 ff.
Hilbert D. & Ackermann W. (1938). Grundzüge der theoretischen Logik (second edition). Berlin: Springer.
Hilbert D. & Bernays P. (1934). Grundlagen der Mathematik (first edition), Vol. I. Berlin: Springer.
Hilbert D. & Bernays P. (1939). Grundlagen der Mathematik (first edition), Vol. II. Berlin: Springer.
Hinkis A. (2013). Proofs of the Cantor-Bernstein Theorem: A Mathematical Excursion. Science Networks. Historical Studies, Vol. 45. Heidelberg: Birkhäuser/Springer.
Hölder O. (1926). Der angebliche circulus vitiosus und die sogenannte Grundlagenkrise in der Analysis. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaft zu Leipzig, Philologisch-Historische Klasse, 78, 243250.
Jech T. (2003). Set Theory. Springer Monographs in Mathematics. Berlin: Springer.
Jockusch C. G. Jr. & Soare R. I. (1972). ${\rm{\Pi }}_1^0 $ classes and degrees of theories. Transactions of the American Mathematical Society, 173, 3356.
Kanovei V. G. & Lyubetskii V. A. (2003). On some classical problems of descriptive set theory. Russian Mathematical Surveys, 58(5), 839927.
Kechris A. S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics, Vol. 156. New York: Springer.
Keldysh L. (1974). The ideas of N.N. Luzin in descriptive set theory. Russian Mathematical Surveys, 29(5), 179193.
Kleene S. C. (1952a). Introduction to Metamathematics. Bibliotheca Mathematica, Vol. 1. Amsterdam: North-Holland.
Kleene S. C. (1952b). Recursive functions and intuitionistic mathematics. In Graves L., Hille E., Smith P., and Zariski O., editors. Proceedings of the International Congress of Mathematicians, 1950. Providence: American Mathematical Society, pp. 679685.
Kleene S. C. (1955). Hierarchies of number-theoretic predicates. Bulletin of the American Mathematical Society, 61, 193213.
Kleene S. C. (1959). Quantification of number-theoretic functions. Compositio Mathematica, 14, 2340.
Koellner P. (2009). Truth in mathematics: The question of pluralism. In Bueno O. and Linnebo O., editors. New Waves in the Philosophy of Mathematics. New York: Palmgrave, pp. 80116.
Kohlenbach U. (2008). Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Berlin: Springer.
Kondô M. (1939). Sur l’uniformisation des complémentaires analytiques et les ensembles projectifs de la seconde classe. Japanese Journal of Mathematics, 15, 197230.
Kondô M. (1956). Sur la nommabilité d’ensembles. Comptes Rendus Hebdomadaires des séances de l’Académie des Sciences, 242, 18411843.
Kondô M. (1958). Sur les ensembles nommables et le fondement de l’analyse mathématique. I. Japanese Journal of Mathematics, 28, 1116.
Kondô M. (1960). Le fondement constructif du calcul infinitésimal. Osaka Journal of Mathematics, 12, 6196.
Kondô M. (1985). In Tugué T., Tanaka H., and Yasuda Y., editors. Selected Works of Motokiti Kondô. Tokyo: Shige Kondô.
König D. (1927). Über eine Schlussweise aus dem Endlichen ins Unendliche. Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, 3, 121130.
König D. (1936). Theorie der endlichen und unendlichen Graphen. Leipzig: Akademische Verlagsgesellschaft.
König D. (1990). Theory of Finite and Infinite Graphs. New York: Springer.
Krajewski S. & Woleriski J. (2007). Andrzej Grzegorczyk: Logic and philosophy. Topics in Logic, Philosophy and Foundations of Mathematics, and Computer Science: In Recognition of Professor Andrzej Grzegorczyk, 81, 117.
Kreisel G. (1950). Note on arithmetic models for consistent formulae of the predicate calculus. Fundamenta Mathematicae, 37, 265285.
Kreisel G. (1953). A variant to Hilbert’s theory of the foundations of arithmetic. The British Journal for the Philosophy of Science, 4(14), 107129.
Kreisel G. (1955). Review of kleene (1955). Mathematical Reviews. MR0070593 (17,4f).
Kreisel G. (1958a). Elementary completeness properties of intuitionistic logic with a note on negations of prenex formulae. The Journal of Symbolic Logic, 23(3), 317330.
Kreisel G. (1958b). Mathematical significance of consistency proofs. The Journal of Symbolic Logic, 23(2), 155182.
Kreisel G. (1959a). Analysis of the Cantor-Bendixson theorem by means of the analytic hierarchy. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 7, 621626.
Kreisel G. (1959b). Review of Kondô (1958). Mathematical Reviews. MR0113806 (22 #4638).
Kreisel G. (1960a). La prédicativité. Bulletin de la Société Mathématique de France, 88, 371391.
Kreisel G. (1960b). Ordinal logics and the characterization of informal concepts of proof. In Todd J. A., editor. Proceedings of the International Congress of Mathematicians, 1958. New York: Cambridge University Press, pp. 289299.
Kreisel G. (1962a). The axiom of choice and the class of hyperarithmetic functions. Indagationes Mathematicae, 24, 307319.
Kreisel G. (1962b). On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic, 27, 139158.
Kreisel G. (1967). Informal rigour and completeness proofs [with discussion]. In Lakatos I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138186.
Kreisel G. (1968). A survey of proof theory. The Journal of Symbolic Logic, 33, 321388.
Kreisel G. (1970a). Church’s Thesis: A kind of reducibility axiom for constructive mathematics. In Kino A., Myhill J., and Vesley R. E., editors. Intuitionism and Proof Theory. Amsterdam: North-Holland, pp. 121150.
Kreisel G. (1970b). Principles of proof and ordinals implicit in given concepts. In Kino A., Myhill J., and Vesley R. E., editors. Intuitionism and Proof Theory. Amsterdam: North-Holland, pp. 489516.
Kreisel G. (1976). What have we learnt from Hilbert’s second problem? In Browder F. E., editor. Mathematical Developments Arising from Hilbert Problems. Providence: American Mathematical Society, pp. 93130.
Kreisel G., Mints G., & Simpson S. G. (1975). The use of abstract language in elementary metamathematics: Some pedagogic examples. In Logic Colloquium. Berlin: Springer, pp. 38131.
Kuratowski C. (1922). Une méthode d’élimination des nombres transfinis des raisonnements mathématiques. Fundamenta Mathematicae, 3(1), 76108.
Lebesgue H. (1905). Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées 6e série, 1, 139216.
Leisenring A. C. (1969). Mathematical Logic and Hilbert’s ε-Symbol. New York: Gordon and Breach.
Lusin N. (1925). Sur le probleme de M. Émile Borel et la méthode des résolvantes. Comptes rendus hebdomadaires des séances de l’Académie des Sciences, 181, 279281.
Lusin N. (1930a). Analogies entre les ensembles mesurables B et les ensembles analytiques. Fundamenta Mathematicae, 1(16), 4876.
Lusin N. (1930b). Leçons sur les ensembles analytiques et leurs applications. Collection de monographies sur la théorie des fonctions. Paris: Gauthier-Villars.
Mancosu P. (2003). The Russellian influence on Hilbert and his school. Synthese, 137, 59101.
Michel A. (2008). Remarks on the supposed French ‘semi-’ or ‘pre-intuitionism’. In van Atten M., Boldini P., Bourdeau M., and Heinzmann G., editors. One Hundred Years of Intuitionism (1907–2007). The Cerisy Conference. Publications des Archives Henri-Poincaré. Basel: Birkhäuser, pp. 149162.
Montalbán A. & Shore R. A. (2012). The limits of determinacy in second-order arithmetic. Proceedings of the London Mathematical Society, 104(2), 223252.
Moore G. H. (1982). Zermelo’s Axiom of Choice. Studies in the History of Mathematics and Physical Sciences, Vol. 8. New York: Springer.
Moore G. H. & Garciadiego A. (1981). Burali-Forti’s paradox: A reappraisal of its origins. Historia Mathematica, 8(3), 319350.
Moschovakis Y. N. (1980). Descriptive Set Theory. Studies in Logic and the Foundations of Mathematics, Vol. 100. Amsterdam: North-Holland.
Moschovakis Y. N. (2009). Descriptive Set Theory (second edition). Mathematical Surveys and Monographs, Vol. 155. Providence: American Mathematical Society.
Mostowski A. (1950). Some impredicative definitions in the axiomatic set-theory. Fundamenta Mathematicae, 37(1), 110124.
Mostowski A. (1956). On models of axiomatic set theory. Bulletin de l’Académie Polonaise des Sciences, 4, 663667.
Mostowski A. (1959). On various degrees of constructivism. In Heyting A., editor. Constructivity in Mathematics, Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, pp. 178194.
Parsons C. (1970). On a number theoretic choice schema and its relation to induction. Studies in Logic and the Foundations of Mathematics, 60, 459473.
Parsons C. (2002). Realism and the debate on impredicativity, 1917–1944. In Feferman S., Sieg W., Sommer R., and Talcott C. L., editors. Reflections on the Foundations of Mathematics (Stanford, CA, 1998). Lecture Notes in Logic, Vol. 15. Urbana: Associaton of Symbolic Logic, pp. 372389.
Pohlers W. (1987). Contributions of the Schütte school. In Takeuti G., editor. Proof Theory, Second Edition. Studies in Logic and the Foundations of Mathematics, Vol. 81. Amsterdam: North-Holland, pp. 406431.
Poincaré H. (1905). Les mathématiques et la logique. Revue de métaphysique et de morale, 13, 815835.
Poincaré H. (1906). Les mathématiques et la logique. Revue de métaphysique et de morale, 14, 294317.
Poincaré H. (1909a). La logique de l’infini. Revue de métaphysique et de morale, 17, 461482.
Poincaré H. (1909b). Réflexions sur les deux notes précédentes. Acta Mathematica, 32, 195200.
Poincaré H. (1910). Über transfinite Zahlen. In Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathematischen Physik. Mathematische Vorlesungen an der Universität Göttingen. Leipzig/Berlin: Teubner, pp. 4448.
Post E. L. (1948). Degrees of unsolvability – preliminary report. Bulletin of the American Mathematical Society, 54(7), 641642.
Ramsey F. P. (1925). The foundations of mathematics. Proceedings of the London Mathematical Society, 25(5), 338384.
Ramsey F. P. (1926). Mathematical logic. The Mathematical Gazette, 13(184), 185194.
Rosser J. B. & Wang H. (1950). Non-standard models for formal logics. The Journal of Symbolic Logic, 15(02), 113129.
Rubin H. & Rubin J. E. (1963). Equivalents of the Axiom of Choice, Vols. I,II. Amsterdam: North-Holland.
Russell B. (1906). Les paradoxes de la logique. Revue de métaphysique et de morale, 14(5), 627650.
Russell B. (1907). On some difficulities in the theory of transfinite numbers and order types. Proceedings of the London Mathematical Society, 2–4(1), 2953. This is text of a talk from December 14, 1905.
Russell B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222262.
Russell B. (1910). La théorie des types logiques. Revue de Métaphysique et de Morale, 18(3), 263301.
Sacks G. E. (1990). Higher Recursion Theory. Perspectives in Mathematical Logic. Berlin: Springer.
Schütte K. (1960). Beweistheorie. Die Grundlehren der mathematischen Wissenschaften, Vol. 103. Berlin: Springer.
Schütte K. (1965a). Eine Grenze für die Beweisbarkeit der transfiniten Induktion in der verzweigten Typenlogik. Archiv für Mathematische Logik und Grundlagenforschung, 7, 4560 (1965).
Schütte K. (1965b). Predicative well-orderings. In Crossley J. N. and Dummett M., editors. Formal Systems and Recursive Functions. Amsterdam: North-Holland, pp. 280303.
Schütte K. (1977). Proof Theory. Grundlehren der Mathematischen Wissenschaften, Vol. 225. Berlin: Springer.
Scott D. (1962). Algebras of sets binumerable in complete extensions of arithmetic. In Dekker J. C. E., editor. Proceedings of Symposia in Pure Mathematics, Vol. 5. Providence: American Mathematical Society, pp. 117121.
Shoenfield J. R. (1960). Degrees of models. The Journal of Symbolic Logic, 25(3), 233237.
Shoenfield J. R. (1967). Mathematical Logic, Vol. 21. Reading: Addison-Wesley.
Shore R. A. (2010). Reverse mathematics: The playground of logic. The Bulletin of Symbolic Logic, 16(3), 378402.
Sieg W. (1985). Fragments of arithmetic. Annals of Pure and Applied Logic, 28(1), 3371.
Sieg W. (2009). Hilbert’s proof theory. In Gabbay D. M. and Woods J., editors. Handbook of the History of Logic. Volume 5: Logic from Russell to Church. Amsterdam: North-Holland, pp. 321384.
Sieg W. & Schlimm D. (2005). Dedekind’s analysis of number: Systems and axioms. Synthese, 147(1), 121170.
Simpson S. G. (1973). Notes on subsystems of analysis. Unpublished, typewritten, Berkeley, 38 pages.
Simpson S. G. (1985). Friedman’s research on subsystems of second order arithmetic. In Harrington L. A., Morley M. D., Scedrov A., and Simpson S. G., editors. Harvey Friedman’s Research on the Foundations of Mathematics. Studies in Logic and the Foundations of Mathematics, Vol. 117. Amsterdam: North-Holland, pp. 137159.
Simpson S. G. (1988). Partial realizations of Hilbert’s program. The Journal of Symbolic Logic, 53(2), 349363.
Simpson S. G. (1999). Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Berlin: Springer.
Simpson S. G. (2009). Subsystems of Second Order Arithmetic (second edition). Perspectives in Mathematical Logic. Cambridge: Cambridge University Press.
Souslin M. (1917). Sur un définition des ensembles measurables B sans nombres transfinis. Comptes rendus hebdomadaires des séances de l’Académie des Sciences, 164, 8891.
Steel J. (1975). Descending sequences of degrees. The Journal of Symbolic Logic, 40(01), 5961.
Steel J. (1977). Determinateness and subsystems of analysis.Dissertation, University of California, Berkeley, Unpublished.
Sullivan P. M. (2004). Frege’s logic. In Gabbay D. M. and Woods J., editors. The Rise of Modern Logic: From Leibniz to Frege. Handbook of the History of Logic, Vol. 3. Amsterdam: Elsevier/North-Holland, pp. 659750.
Tait W. W. (1968). Constructive reasoning. In van Rootselaar B. and Staal J. F., editors. Logic, Methodology and Philosophy of Science III. Amsterdam: North-Holland, pp. 185199.
Tait W. W. (1981). Finitism. The Journal of Philosophy, 78(9), 524546.
Tarski A. (1936). Der Wahrheitsbegriff in den formalisierten Sprache. Studia Philosophica, 1, 261405.
Tarski A. (1956). Logic, Semantics, and Metamathematics. Oxford: Clarendon Press.
Troelstra A. & van Dalen D. (1988). Constructivism in Mathematics, An Introduction, Vol. 2. Amsterdam: North-Holland.
Troelstra A. S. (1982). On the origin and developement of Brouwer’s concept of choice sequence. In Troelstra A. S. and van Dalen D., editors. The L. E. J. Brouwer Centenary Symposium. Studies in Logic and the Foundations of Mathematics, Vol. 110. Amtersdam: North-Holland, pp. 465477.
Urquhart A. (2003). The theory of types. In Griffin N., editor. The Cambridge Companion to Russell. Cambridge: Cambridge University Press, pp. 286309.
van Dalen D. (1973). Lectures on intuitionism. In Mathias A. and Rogers H., editors. Cambridge Summer School in Mathematical Logic. Berlin: Springer, pp. 194.
van Heijenoort J., editor (1967). From Frege to Gödel : A Source Book in Mathematical Logic, 1879–1931. Cambridge: Harvard University Press.
Veldman W. (2014). Brouwer’s fan theorem as an axiom and as a contrast to Kleene’s alternative. Archive for Mathematical Logic, 53(5–6), 621693.
Von Neumann J. (1925). Eine Axiomatisierung der Mengenlehre. Journal für die reine und angewandte Mathematik, 154, 219240.
Walsh S. (2014). Logicism, interpretability, and knowledge of arithmetic. The Review of Symbolic Logic, 7(1), 84119.
Walsh S. (2016). Predicativity, the Russell-Myhill paradox, and Church’s intensional logic. The Journal of Philosophical Logic, 45(3), 277326.
Wang H. (1953). Between number theory and set theory. Mathematische Annalen, 126(1), 385409.
Wang H. (1954). The formalization of mathematics. The Journal of Symbolic Logic, 19, 241266.
Wang H. (1955). On denumerable bases of formal systems. In Skolem T., editor. Mathematical Interpretation of Formal Systems. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, pp. 5784.
Wang H. (1974). From Mathematics to Philosophy. New York: Humanities Press.
Weyl H. (1910). Über die Definitionen der mathematischen Grundbegriffe. Mathematisch-naturwissenschaftliche Blätter, 7, 9395.
Weyl H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Leipzig: Veit.
Weyl H. (1921). Über die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift, 10, 3979.
Weyl H. (1926). Die heutige Erkenntnislage in der Mathematik. Symposion; philosophische Zeitschrift für Forschung und Ausspräche, 1, 132.
Weyl H. (1968). In Chandrasekharan K., editor. Gesammelte Abhandlungen. New York: Springer.
Whitehead A. N. & Russell B. (1910). Principia Mathematica. Cambridge: Cambridge University Press.
Whitehead A. N. & Russell B. (1962). Principia Mathematica to *56. Cambridge: Cambridge University Press.
Zach R. (2003). The practice of finitism: Epsilon calculus and consistency proofs in Hilbert’s program. Synthese, 137(1–2), 211259.
Zermelo E. (1908a). Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, 65(1), 107128.
Zermelo E. (1908b). Untersuchungen über die Grundlagen der Mengenlehre. I. Mathematische Annalen, 65(2), 261281.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 53 *
Loading metrics...

Abstract views

Total abstract views: 345 *
Loading metrics...

* Views captured on Cambridge Core between 20th February 2017 - 21st November 2017. This data will be updated every 24 hours.