Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T07:16:24.620Z Has data issue: false hasContentIssue false
  • English
  • Français

Undecidability of the real-algebraic structure of models of intuitionistic elementary analysis

Published online by Cambridge University Press:  12 March 2014

Miklós Erdélyi-Szabó
Miklós Erdélyi-Szabó*
Affiliation:
Mindmaker Ltd. H1121 Budapest, Konkoly Thege Mikclós Út 29–33, Hungary E-mail: mszabo@mindmaker.hu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that true first-order arithmetic is interpretable over the real-algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras. From this the undecidability of the structures follows. We also show that Scott's model is equivalent to true second-order arithmetic. In the appendix we argue that undecidability on the language of ordered rings follows from intuitionistically plausible properties of the real numbers.

Keywords

UndecidabilityIntuitionismHeyting algebraTrue second-order arithmetic
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1014 - 1030
Copyright
Copyright © Association for Symbolic Logic 2000

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]Cherlin, G., Rings of continuous functions: Decision problems, Model Theory of Algebra and Arithmetic (Dold, A. and Eckmann, B., editors), Lecture Notes in Mathematics 834, Springer-Verlag, Berlin, Heidelberg, New York, 1980, pp. 44–91.Google Scholar
[2]Dragalin, A.G., Mathematical Intuitionism: Introduction to Proof Theory, AMS, Providence, 1987.Google Scholar
[3]Erdélyi-Szabó, M., Undecidability of the real-algebraic structure of Scott's model, Mathematical Logic Quarterly, vol. 44 (1998), pp. 344–348.CrossRefGoogle Scholar
[4]Heyting, A., Intuitionism. An Introduction, North-Holland, Amsterdam, 1956.Google Scholar
[5]Kleene, S.C. and Vesley, R.E., The Foundations of Intuitionistic Mathematics, North-Holland, Amsterdam, 1965.Google Scholar
[6]Krol, M.D., A topological model for intuitionistic analysis with Kripke's scheme, Z. Math. Logik Grundlag. Math., vol. 24 (1978), pp. 427–436.CrossRefGoogle Scholar
[7]Scott, D.S., Extending the topological interpretation to intuitionistic analysis, Compositio Math., vol. 20 (1968), pp. 194–210.Google Scholar
[8]Scowcroft, P., Some purely topological models for intuitionistic analysis, submitted to Annals of Pure and Applied Logic.Google Scholar
[9]Scowcroft, P., A new model for intuitionistic analysis, Annals of Pure and Applied Logic, vol. 47 (1990), pp. 145–165.CrossRefGoogle Scholar
[10]van Dalen, D., Intuitionistic logic, Handbook of Philosophical Logic, Vol. III (Gabbay, D. and Guenthner, F., editors), D. Reidel Publishing Company, 1986, pp. 225–339.Google Scholar