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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

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