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Degrees of formal systems1

Published online by Cambridge University Press:  12 March 2014

J. R. Shoenfield
J. R. Shoenfield*
Affiliation:
Duke University
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Extract

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Gödel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.

In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.

Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 4 , December 1958 , pp. 389 - 392
Copyright
Copyright © Association for Symbolic Logic 1958

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Footnotes

1

Presented to the American Mathematical Society January 29, 1958. This research was supported in part by the National Science Foundation of the United States.

References

REFERENCES

[1]Craig, W., On axiomatizability within a system, this Journal, vol. 18 (1953), pp. 30–32.Google Scholar
[2]Feferman, S., Degrees of unsolvability associated with classes of formalized theories, this Journal, vol. 22 (1957), pp. 161–175.Google Scholar
[3]Janiczak, A., Undeciddbility of some simple formalized theories, Fundamenta mathematicae, vol. 40 (1953), pp. 131–139.CrossRefGoogle Scholar
[4]Kleene, S. C., Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952.Google Scholar
[5]Rogers, Hartley Jr., Theory of recursive functions and effective computability, vol. 1, mimeographed, Cambridge, Mass., 1957.Google Scholar