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Constructive set theory

Published online by Cambridge University Press:  12 March 2014

John Myhill
John Myhill*
Affiliation:
Suny at Buffalo, Amherst, New York 14226
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Extract

This paper is the third in a series collectively entitled Formal systems of intuitionistic analysis. The first two are [4] and [5] in the bibliography; in them I attempted to codify Brouwer's mathematical practice. In the present paper, which is independent of [4] and [5], I shall do the same for Bishop's book [1]. There is a widespread current impression, due partly to Bishop himself (see [2]) and partly to Goodman and the author (see [3]) that the theory of Gödel functionals, with quantifiers and choice, is the appropriate formalism for [1]. That this is not so is seen as soon as one really tries to formalize the mathematics of [1] in detail. Even so simple a matter as the definition of the partial function 1/x on the nonzero reals is quite a headache, unless one is prepared either to distinguish nonzero reals from reals (a nonzero real being a pair consisting of a real x and an integer n with ∣x∣ > 1/n) or, to take the Dialectica interpretation seriously, by adjoining to the Gödel system an axiom saying that every formula is equivalent to its Dialectica interpretation. (See [1, p. 19], [2, pp. 57–60] respectively for these two methods.) In more advanced mathematics the complexities become intolerable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 3 , September 1975 , pp. 347 - 382
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[1]Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[2]Bishop, E., Mathematics as a numerical language, Intuitionism and proof theory (Kino, , Myhill, and Vesley, , Editors), North-Holland, Amsterdam, 1970, pp. 53–71.Google Scholar
[3]Goodman, N. and Myhill, J., The formalization of Bishop's constructive mathematics, Toposes, algebraic geometry, and logic (Lawvere, , Editor), Springer, Berlin, 1972, pp. 83–96.CrossRefGoogle Scholar
[4]Myhill, J., Formal systems of intuitionistic analysis. I, Logic, methodology and philosophy of science. III (Rootselaar, van and Staal, , Editors), North-Holland, Amsterdam, 1968, pp. 161–178.CrossRefGoogle Scholar
[5]Myhill, J., Formal systems of intuitionistic analysis. II, Intuitionism and proof theory (Kino, , Myhill, and Vesley, , Editors), North-Holland, Amsterdam, 1970, pp. 151–162.Google Scholar
[6]Myhill, J., Some properties of intuitionistic Zermelo-Frankel set theory, Proceedings of the Summer Logic Conference at Cambridge, 08 1971, Springer, Berlin, pp. 206–231.Google Scholar
[7]Myhill, J. and Goodman, N., The axiom of choice and the law of excluded middle (to appear).Google Scholar