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  • Cited by 10
  • Print publication year: 2002
  • Online publication date: March 2017

Foundational and mathematical uses of higher types

[1] Avigad, J., Feferman, S., Gödel's functional (‘Dialectica’) interpretation. In: [3], pp. 337-405.(1998).
[2] Beeson, M.J., Foundations of ConstructiveMathematics. Springer Ergebnisse derMathematik und ihrer Grenzgebiete 3.Folge, Bd.6., Berlin Heidelberg New York Tokyo (1985).
[3] Buss, S.R. (editor), Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics Vol 137, Elsevier, vii+811 pp. (1998).
[4] Feferman, S., Alanguage and axioms for explicit mathematics. In: Crossley, J.N. (ed.), Algebra and Logic, pp. 87-139. Springer Lecture Notes in Mathematics 450 (1975).
[5] Feferman, S., Theories of finite type related to mathematical practice. In: Barwise, J.|(ed.), Handbook of Mathematical Logic, pp. 913-972. North-Holland, Amsterdam (1977).
[6] Feferman, S., Working foundations. Synthese 62, pp. 229-254.(1985).
[7] Feferman, S., Weyl vindicated: Das Kontinuum seventy years later, in: Cellucci, C., Sambin, G.|(eds.), Temi e prospettive della logica e della filosofia della scienza contemporance, vol. I, pp. 59-93.(1988), CLUEB, Bologna. Reprinted (with minor additions) in [10].
[8] Feferman, S., Infinity in mathematics: Is Cantor necessary?. In: G., Toraldo di Francia (ed.), L'infinito nella scienza, Istituto della Enciclopedia Italiana, Rome, pp. 151-209.(1987). Reprinted (with minor additions) in [10].
[9] Feferman, S., Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics. In PSA (1992), Vol.2 (Philosophy of Science Association, East Lansing), pp. 442-455. Reprinted (with minor additions) in [10].
[10] Feferman, S., In the Light of Logic. Oxford University Press, 340 pp. (1998).
[11] Feferman, S., Jäger, G., Systems of explicit mathematics with non-constructive _-operator. Ann. Pure Applied Logic, Part I. 65, pp. 243-263.(1993), Part II. 79, pp. 37-52.(1996).
[12] Felgner, U., Models of ZF-Set Theory. Lecture Notes in Mathematics 223, Springer, Berlin (1971).
[13] Gödel, K., Über eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280-287.(1958).
[14] Grilliot, T.J., On effectively discontinuous type-2 objects. J. Symbolic Logic 36, pp. 245-248.(1971).
[14] Grilliot, T.J., On effectively discontinuous type-2 objects. J. Symbolic Logic 36, pp. 245-248 (1971).
[15] Halpern|J.D.-Levy, A., The Boolean Prime Ideal theorem does not imply the axiom of choice. In: Proc. of Symp. Pure Math. vol.XIII, pp.83-134, AMS, Providence (1971).
[16] Jägermann, M., The axiom of choice and two definitions of continuity. Bull. Acad. Polon. Sci. 13, pp.699-704 (1965).
[17] Kohlenbach, U., Remarks on Herbrand normal forms and Herbrand realizations. Arch. Math. Logic 31, pp. 305-317.(1992).
[18] Kohlenbach, U., Pointwise hereditary majorization and some applications. Arch.Math. Logic 31, pp. 227-241.(1992).
[19] Kohlenbach, U., Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization. J. Symbolic Logic 57, pp. 1239-1273.(1992).
[20] Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation. Ann. Pure Appl. Logic 64, pp. 27-94.(1993).
[21] Kohlenbach, U., Analysing proofs in analysis. In: W., Hodges, M., Hyland, C., Steinhorn, J., Truss, editors, Logic: from Foundations to Applications. European Logic Colloquium (Keele, 1993), pp. 225-260. Oxford University Press (1996).
[22] Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).
[23] Kohlenbach, U., Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals. Arch. Math. Logic 36, pp. 31-71.(1996).
[24] Kohlenbach, U., Proof theory and computational analysis. Electronic Notes in Theoretical Computer Science 13, Elsevier (, (1998).
[25] Kohlenbach, U., The use of a logical principle of uniformboundedness in analysis. In: Cantini, A., Casari, E., Minari, P. (eds.), Logic and Foundations of Mathematics. Synthese Library 280, pp. 93-106. Kluwer Academic Publishers (1999).
[26] Kohlenbach, U., Arithmetizing proofs in analysis. In: Larrazabal, J.M. et al. (eds.), Proceedings Logic Colloquium 96 (San Sebastian), Springer Lecture Notes in Logic 12, pp. 115-158.(1998).
[27] Kohlenbach, U., On the arithmetical content of restricted forms of comprehension, choice and general uniform boundedness. Ann. Pure and Applied Logic 95, pp. 257-285.(1998).
[28] Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491- 1511.(1999).
[29] Kohlenbach, U., Things that can and things that can't be done in PRA. Ann. Pure Applied Logic 102, pp. 223-245.(2000).
[30] Kohlenbach, U., On the uniform weak König's lemma. To appear in: Annals of Pure Applied Logic.
[31] Kohlenbach, U., Higher order reverse mathematics. Preprint 14pp., submitted.
[32] Normann, D., Recursion on the Countable Functionals. Springer Lecture Notes in Mathematics 811 (1980).
[33] Sieg, W., Fragments of arithmetic. Ann. Pure Appl. Logic 28,pp. 33-71.(1985).
[34] Simpson, S.G., Partial realizations of Hilbert's program. J. Symbolic Logic 53, 349-363 (1988).
[35] Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives inMathematical Logic. Springer-Verlag. xiv+445 pp. (1999).
[36] Tait, W.W., Finitism. Journal of Philosophy 78, pp. 524-546.(1981).
[37] Takeuti, G., A conservative extension of Peano arithmetic. Part II of ‘Two applications of logic to mathematics’, Publ. Math. Soc. Japan 13, (1978).
[38] Troelstra, A.S. (ed.)Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes inMathematics 344 (1973).
[39] Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596.(1974).
[40] Weyl, H., Das, Kontinuum. Kritische Untersuchungenüber die Grundlagen der Analysis. Veit, Leipzig (1918). English translation: Weyl, H., The Continuum: A critical Examination of the Foundation of Analysis. New York, Dover (1994).