Skip to main content Accessibility help
Hostname: page-component-59b7f5684b-b2xwp Total loading time: 0.441 Render date: 2022-10-01T05:26:56.109Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Foundational and mathematical uses of higher types


Published online by Cambridge University Press:  31 March 2017

Wilfried Sieg
Carnegie Mellon University, Pennsylvania
Richard Sommer
Stanford University, California
Carolyn Talcott
Stanford University, California
Get access


Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Reflections on the Foundations of Mathematics
Essays in Honor of Solomon Feferman
, pp. 92 - 116
Publisher: Cambridge University Press
Print publication year: 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[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).Google Scholar
[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).Google Scholar
[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).Google Scholar
[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).Google Scholar
[17] Kohlenbach, U., Remarks on Herbrand normal forms and Herbrand realizations. Arch. Math. Logic 31, pp. 305-317.(1992).Google Scholar
[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).Google Scholar
[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).Google Scholar
[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).Google Scholar
[28] Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491- 1511.(1999).Google Scholar
[29] Kohlenbach, U., Things that can and things that can't be done in PRA. Ann. Pure Applied Logic 102, pp. 223-245.(2000).Google Scholar
[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).Google Scholar
[33] Sieg, W., Fragments of arithmetic. Ann. Pure Appl. Logic 28,pp. 33-71.(1985).Google Scholar
[34] Simpson, S.G., Partial realizations of Hilbert's program. J. Symbolic Logic 53, 349-363 (1988).Google Scholar
[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).Google Scholar
[37] Takeuti, G., A conservative extension of Peano arithmetic. Part II of ‘Two applications of logic to mathematics’, Publ. Math. Soc. Japan 13, (1978).Google Scholar
[38] Troelstra, A.S. (ed.)Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes inMathematics 344 (1973).Google Scholar
[39] Troelstra, A.S., Note on the fan theorem. J. Symbolic Logic 39, pp. 584-596.(1974).Google Scholar
[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).
Cited by

Save book to Kindle

To save this book to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats