Abstract
Church introduced the λ-calculus in the beginning of the thirties as a foundation of mathematics, and map theory, from around 1992, fulfilled that primary aim.
The present paper presents a new version of map theory whose axioms are simpler and better motivated than those of the original version from 1992. The paper focuses on the semantics of map theory and explains this semantics on the basis of κ-Scott domains.
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References
Aczel, P. 1980 Frege structures and the notions of proposition, truth and set, The Kleene symposium, (J. Barwise, H. J. Keisler, and K. Kunen, editors), North-Holland, Amsterdam, pp. 31–59.
Berline, C., and K. Grue 1997 A K-denotational semantics for Map Theory in ZFC+SI, Theoretical Computer Scienc, vol. 179, no. 1-2 (June), pp. 137–202.
Chang, C. C., and K. J. Keisler 1973 Model theory, Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam.
Church, A. 1933 A set of postulates for the foundations of logic I, Annals of Mathematics, vol. 33, pp. 346–366.
Church, A. 1934 A set of postulates for the foundations of logic II, Annals of Mathematics, vol. 34, pp. 839–864.
Church, A. 1941 The calculi of lambda conversion, Princeton University Press, Princeton, New Jersey.
Feferman, S. 1978 Recursion theory and set theory, a marriage of convenience, Generalized recursion theory II, Proceedings of the 1977 Oslo Symposium (J. E. Fenstad, R. O. Gandy, and G. E. Sacks, editors), North-Holland, Amsterdam, pp. 55–98.
Feferman, S. 1984 Toward useful type-free theories I, The Journal of Symbolic Logic, vol. 49, pp. 75–111.
Feigner, U. 1976 Choice functions on sets and classes, Sets and classes: On the works by Paul Bernays, North-Holland, Amsterdam, pp. 217–255.
Flagg, R. C., and J. Myhill 1989 A type-free system extending ZFC, Annals of Pure and Applied Logic, vol. 43, pp. 79–97.
Grue, K. 1992 Map theory, Theoretical Computer Science, vol. 102, no. 1 (July), pp. 1–133.
Grue, K. 1996 Stable map theory, Department of Computer Science, University of Copenhagen, DIKU, Universitetsparken 1, DK-2100 Copenhagen, Denmark”, DIKU Report, no. 96/10 (April); available from //www.diku.dk/~grue.
Hilbert, D., and P. Bernays 1939 Grundlagen der Mathematica vol. 2, Springer-Verlag, Berlin.
Mendelson, E. 1987 Introduction to mathematical logic, third edition, Wadsworth and Brooks, Monterey, California.
Scott, D. 1973 Models for various type-free λ-calculi, Proceedings of the IVth international congress for logic (Suppes et al., editors), Studies in Logic and The Foundation of Mathematics, Methodology and Philosophy of Science, North-Holland, Amsterdam, vol. 74, pp. 157–187.
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Grue, K. (2001). λ-Calculus as a Foundation for Mathematics. In: Anderson, C.A., Zelëny, M. (eds) Logic, Meaning and Computation. Synthese Library, vol 305. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0526-5_13
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DOI: https://doi.org/10.1007/978-94-010-0526-5_13
Publisher Name: Springer, Dordrecht
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