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# A logical presentation of the continuous functionals

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The Kleene-Kreisel continuous functionals [6, 7] have been given several alternative characterisations: using Kuratowski's limit spaces (Scarpellini [20]), using their generalisation, filter spaces (Hyland [4]), via hyperfinite functionals (Normann [11]) and perhaps most elegantly using Scott-Ershov domains (Ershov [3], later generalised by Berger [1]). We propose to add yet another characterisation to this list, which may be called model-theoretic in contrast to the others, but which is in fact closely related to Ershov's approach.

We use the notion of logically presented domains developed in Palmgren and Stoltenberg-Hansen [17] (originating in the work of [15]). Certain logical types, i.e., finitely consistent sets of formulas, over the full type structure built from ℕ correspond to the continuous functionals in the sense of Kreisel, or to Kleene's associates, while the elements realising the types correspond to functionals having associates (cf. [6]). To define these—the total types—we make a nonstandard extension of the full type structure. The set of nonstandard elements is sufficiently rich to single out the total types. The nonstandard extension also makes it possible to relate the logical presentation to Ershov's approach. The logical form of the domain constructions allows us to use a (generalised) Fréchet power as a nonstandard extension. This extension is constructive, in the sense that it avoids the axiom of choice, as distinguished from the one employed in [11].

References
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[1] Berger, U., Total sets and objects in domain theory, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 91117.
[2] Chang, C. C. and Keisler, H. J., Model theory, third ed., North-Holland, Amsterdam, 1990.
[3] Ershov, Yu. L., The model C of the continuous functionals, Logic colloquium '76 (Gandy, R. and Hyland, M., editors), North-Holland, Amsterdam, 1977.
[4] Hyland, J. M. E., Filter spaces and continuous functionals, Annals of Mathematical Logic, vol. 16 (1979), pp. 101143.
[5] Jónsson, B. and Olin, P., Almost direct products and saturation, Compositio Mathematica, vol. 20 (1968), pp. 125132.
[6] Kleene, S. C., Countable functionals, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1959.
[7] Kreisel, G., Interpretation of analysis by means of functionals of finite types I, Constructivity in mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1959.
[8] Lassez, J.-L. and McAloon, K., A constraint sequent calculus, Proceedings of the 5th IEEE symposium on logic in computer science (Washington D.C.), IEEE Computer Society Press, 1990.
[9] Normann, D., The continuous functionals, Handbook of recursion theory (Griffor, E., editor), North-Holland, forthcoming.
[10] Normann, D., Recursion on the continuous functionals, Lecture notes in mathematics, vol. 811, Springer-Verlag, Berlin, 1980.
[11] Normann, D., Characterizing the continuous functionals, this Journal, vol. 48 (1983), pp. 965969.
[12] Normann, D., Formalizing the notion of total information, Mathematical logic (Petkov, P. P., editor), Plenum Press, 1990, pp. 6794.
[13] Normann, D., A hierarchy of domains with totality but without density, Computability, enumerability, unsolvability. Directions in recursion theory (Cooper, S. B.et al., editors), Cambridge University Press, 1996.
[14] Omarov, A. I., A syntactical description of filtering formulas, Soviet Math. Dokl., vol. 44 (1992), pp. 5355.
[15] Palmgren, E., Denotational semantics of constraint logic programming—a nonstandard approach, Constraint programming (Mayoh, B., Tyugu, E., and Penjam, J., editors), NATO ASI Series F, Springer-Verlag, 1994, pp. 261288.
[16] Palmgren, E., A direct proof that certain reduced products are countably saturated, Department of Mathematics Report 27, Uppsala University, 1994.
[17] Palmgren, E. and Stoltenberg-Hansen, V., Logically presented domains, Proceedings of the 10th IEEE symposium on logic in computer science (Washington D.C.), IEEE Computer Society Press, 1995.
[18] Palyutin, E. A., Categorical Horn classes I, Algebra and Logic, vol. 19 (1980), pp. 377400.
[19] Richter, M. M. and Szabo, M. E., Towards a nonstandard analysis of programs, Nonstandard analysis—recent developments (Hurd, A. E., editor), Lecture Notes in Mathematics, vol. 983, Springer-Verlag, Berlin, 1983.
[20] Scarpellini, B., A model for barrecursion of higher types, Compositio Mathematica, vol. 23 (1971), pp. 123153.
[21] Stoltenberg-Hansen, V., Lindström, I., and Griffor, E. R., Mathematical theory of domains, Cambridge University Press, 1994.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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