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The theory of the Gödel functionals

Published online by Cambridge University Press:  12 March 2014

Nicolas D. Goodman
Nicolas D. Goodman*
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
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Extract

In [2] we described an arithmetic theory of constructions (ATC) and showed that first-order intuitionistic arithmetic (HA) could be interpreted in it. In [3] we went on to show that the interpretation of HA in ATC is faithful. The purpose of the present paper is to apply these ideas to intuitionistic arithmetic in all finite types. Tait has shown [6] that a conservative extension of HA is obtained by adding the Gödel functionals with intuitionistic logic and intensional identity in all finite types. Below we show that this extension remains conservative on the addition of certain axioms of choice which are evident on the intended interpretation of the intuitionistic logical connectives. This theorem (Corollary 6.2 below) was first obtained by a more complicated argument in our dissertation [1]. Some of its implications are discussed in Goodman and Myhill [4].

We assume that the reader is familiar with [2] and [3].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 3 , September 1976 , pp. 574 - 582
Copyright
Copyright © Association for Symbolic Logic 1976

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References

REFERENCES

[1] Goodman, N. D., Intuitionistic arithmetic as a theory of constructions, Dissertation, Stanford University, 1968.Google Scholar
[2] Goodman, N. D., The arithmetic theory of constructions, Cambridge Summer School in Mathematical Logic, Proceedings 1971 (edited by Mathias, A. R. D. and Rogers, H.), Springer-Verlag, Berlin, 1973, pp. 273–298.Google Scholar
[3] Goodman, N. D., The faithfulness of the interpretation of arithmetic in the theory of constructions, this Journal, vol. 38 (1973), pp. 453–459.Google Scholar
[4] Goodman, N. D. and Myhill, J., The formalization of Bishop's constructive mathematics, Toposes, algebraic geometry, and logic (edited by Lawvere, F. W.), Springer–Verlag, Berlin, 1972, pp. 83–96.CrossRefGoogle Scholar
[5] Platek, R. A., Foundations of recursion theory, Dissertation, Stanford University, 1966.Google Scholar
[6] Tait, W. W., Intensional interpretation of functionals of finite type. I, this Journal, vol. 32 (1967), pp. 198–212.Google Scholar