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Effectively retractable theories and degrees of undecidability1

Published online by Cambridge University Press:  12 March 2014

J. P. Jones
J. P. Jones*
Affiliation:
The University of Calgary
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Extract

In this paper a new property of theories, called effective retractability is introduced and used to obtain a characterization for the degrees of subtheories of arithmetic and set theory. By theory we understand theory in standard formalization as defined by Tarski [10]. The word degree refers to the Kleene-Post notion of degree of recursive unsolvability [2]. By the degree of a theory we mean, of course, the degree associated with its decision problem via Gödel numbering.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 4 , February 1970 , pp. 597 - 604
Copyright
Copyright © Association for Symbolic Logic 1970

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Footnotes

1

The results in this paper constitute a part of the author's Dissertation written while the author was a graduate student at the University of Washington. The author wishes to express his thanks to Professors R. W. Ritchie of the University of Washington and M. B. Poúr-El of the University of Minnesota for many helpful suggestions regarding the writing of this paper.

References

[1] Feferman, S., Degrees of unsolvability associated with classes of formalized theories, this Journal , vol. 22 (1957), pp. 161–175.Google Scholar
[2] Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, 1952.Google Scholar
[3] Poúr-El, M. B., Effectively extensible theories, this Journal , vol. 33 (1968), pp. 56–68.Google Scholar
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[10] Tarski, A., Mostowski, A. and Robinson, R., Undecidable theories, North-Holland, Amsterdam, 1953.Google Scholar