Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T01:19:02.581Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperfinite type structures

Published online by Cambridge University Press:  12 March 2014

Dag Normann ,
Erik Palmgren  and
Viggo Stoltenberg-Hansen
Dag Normann
Affiliation:
Institute of Mathematics, University of Oslo, Box 1053 Blindern, N-0316 OSLO, Norway E-mail: dnormann@math.uio.no
Erik Palmgren
Affiliation:
Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden E-mail: palmgren@math.uu.se
Viggo Stoltenberg-Hansen
Affiliation:
Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden E-mail: viggo@math.uu.se
Get access Rights & Permissions [Opens in a new window]

Extract

The notion of a hyperfinite set comes from nonstandard analysis. Such a set has the internal cardinality of a nonstandard natural number. By a transfer principle such sets share many properties of finite sets. Here we apply this notion to give a hyperfinite model of the Kleene-Kreisel continuous functionals. We also extend the method to provide a hyperfinite characterisation of certain transfinite type structures, thus, through the work of Waagbø [14], constructing a hyperfinite model for Martin-Löf type theory.

This kind of application is not new. Normann [6] gave a characterisation of the Kleene-Kreisel continuous functionals using ‘hyperfinitary’ functionals. The novelty here is that we use a constructive version of hyperfinite functionals and also generalise the method to transfinite types. Many of the results of this paper are constructive, though not the characterisation theorems themselves.

Our characterisation of the Kleene-Kreisel continuous functionals is a supplement to a number of previous characterisations of topological and recursion-theoretical nature, see [6] for a brief survey. Altogether these characterisations show that the original concept of Kleene and Kreisel forms the correct mathematical model of the idea of finitely based functions of finite types.

There is, however, no a priori reason to believe that there is a canonical way to extend the continuous functionals to cover transfinite objects of transfinite type used in, e.g., type theory. Our characterisation of Waagbø's model indicates that the model is natural, not only seen from domain theory but from a higher perspective. Normann and Waagbø (unpublished) have subsequently obtained a limit-space characterisation that further supports this view.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1216 - 1242
Copyright
Copyright © Association for Symbolic Logic 1999

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.)

References

REFERENCES

[1] Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., North-Holland, Amsterdam, 1990.Google Scholar
[2] Ershov, Yu. L., The model C of the continuous functional, Logic colloquium '76 (Gandy, R. O. and Hyland, J. M. E., editors), North-Holland, Amsterdam, 1977, pp. 455–467.Google Scholar
[3] Jónsson, B. and Olin, P., Almost direct products and saturation, Compositio Mathematica, vol. 20 (1968), pp. 125–132.Google Scholar
[4] Longo, G. and Moggi, E., The hereditarily partial effective functionals and recursion theory in higher types, this Journal, vol. 49 (1984), pp. 1319–1332.Google Scholar
[5] Normann, D., The continuous functionals, Handbook of computability theory (Griffor, E. R., editor), North-Holland, Amsterdam, forthcoming.Google Scholar
[6] Normann, D., Characterizing the continuous functionals, this Journal, vol. 48 (1983), pp. 965–969.Google Scholar
[7] 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, Cambridge, 1996.Google Scholar
[8] Palmgren, E., An information system interpretation of Martin-Löf's partial type theory with universes, Information and Computation, vol. 106 (1993), pp. 26–60.Google Scholar
[9] Palmgren, E. and Stoltenberg-Hansen, V., Domain interpretations of Martin-Löf's partial type theory, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 135–196.Google Scholar
[10] Palmgren, E. and Stoltenberg-Hansen, V., Logically presented domains, Proceedings of the 10th IEEE symposium on logic in computer science, IEEE Computer Society Press, Washington, D.C., 1995, pp. 455–463.Google Scholar
[11] Palmgren, E. and Stoltenberg-Hansen, V., A logical presentation of the continuous functionals, this Journal, vol. 62 (1997), pp. 1021–1034.Google Scholar
[12] Palyutin, E. A., Categorical Horn classes I, Algebra and Logic, vol. 19 (1980), pp. 377–400.Google Scholar
[13] Stoltenberg-Hansen, V., Lindström, I., and Griffor, E. R., Mathematical theory of domains, Cambridge Tracts in Theoretical Computer Science, no. 22, Cambridge University Press, Cambridge, 1994.Google Scholar
[14] Waagbø, G., Denotational semantics for intuitionistic type theory using a hierarchy of domains with totality, Archive for Mathematical Logic, vol. 38 (1999), pp. 19–60.Google Scholar