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Σ2 -collection and the infinite injury priority method

Published online by Cambridge University Press:  12 March 2014

Michael E. Mytilinaios  and
Theodore A. Slaman
Michael E. Mytilinaios
Affiliation:
Department of Mathematics and Computer Science, Dartmouth College Hanover, New Hampshire 03755
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
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Abstract

We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P −) together with Σ2 -collection (BΣ2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor high n-1, while true, cannot be established in P− + BΣn+1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 212 - 221
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1] Groszek, M. J. and Mytilinaios, M. E., Σ2 -induction and the construction of a high degree(to appear).Google Scholar
[2] Groszek, M. J. and Slaman, T. A., Foundations of the priority method. I: Finite and infinite injury (to appear).Google Scholar
[3] Jockusch, C. G. and Shore, R. A., Pseudo jump operators. I: The r. e. case, Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599–609.Google Scholar
[4] Kirby, L. A. S., Initial segments of models of arithmetic, Ph.D. thesis, University of Manchester, Manchester, 1977.Google Scholar
[5] Kirby, L. A. S. and Paris, J. B., Σ2-collection schemas in arithmetic, Logic Colloquium ‘77, North-Holland, Amsterdam, 1978, pp. 199–209.Google Scholar
[6] Mytilinaios, M. E., Finite injury and Σ2-induction (to appear).Google Scholar
[7] Sacks, G. E., Recursive enumerability and the jump operator, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 223–239.CrossRefGoogle Scholar
[8] Shoenfield, J. R., Undecidable and creative theories, Fundamenta Mathematicae, vol. 49 (1961), pp. 171–179.CrossRefGoogle Scholar
[9] Shore, R. A., On the jump of an α-recursively enumerable set, Transactions of the American Mathematical Society, vol. 217 (1976), pp. 351–363.Google Scholar
[10] Slaman, T. A. and Woodin, W. H., Σ1-collection and the finite injury priority method (to appear).Google Scholar
[11] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1986.Google Scholar