Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-04T09:12:46.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizing the continuous functionals

Published online by Cambridge University Press:  12 March 2014

Dag Normann
Dag Normann*
Affiliation:
Matematisk Institutt, Oslo, Norway
Get access Rights & Permissions [Opens in a new window]

Extract

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.

Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.

So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.

In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 4 , December 1983 , pp. 965 - 969
Copyright
Copyright © Association for Symbolic Logic 1983

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]Bergstra, J.A., Computability and continuity in finite types, Thesis, University of Utrecht, 1976.Google Scholar
[2]Ershov, Yu. L., Maximal and everywhere defined functionals, Algebra and Logic, vol. 13 (1974), pp. 210–225.CrossRefGoogle Scholar
[3]Hyland, J.M.E., Filter spaces and continuous functionals, Annals of Mathematical Logic, vol. 16 (1979), pp. 101–143.CrossRefGoogle Scholar
[4]Kleene, S.C., Countable functionals, Constructivity in Mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 81–100.Google Scholar
[5]Kreisel, G., Interpretation of analysis by means of functionals of finite type, Constructivity in Mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 101–128.Google Scholar
[6]Moldestad, J., Partial monotone operators with a countable basis (in preparation).Google Scholar