Journal of Symbolic Logic

$\Sigma_2$-Collection and the Infinite Injury Priority Method

Michael E. Mytilinaios and Theodore A. Slaman

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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 $\Sigma_2$-collection $(B\Sigma_2)$. 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 low$_n$ nor high$_{n - 1}$, while true, cannot be established in $P^- + B\Sigma_{n + 1}$. Consequently, no bounded fragment of first order arithmetic establishes the facts that the high$_n$ and low$_n$ jump hierarchies are proper on the recursively enumerable degrees.

Article information

Source
J. Symbolic Logic, Volume 53, Issue 1 (1988), 212-221.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183742576

Mathematical Reviews number (MathSciNet)
MR929386

Zentralblatt MATH identifier
0645.03039

JSTOR
links.jstor.org

Citation

Mytilinaios, Michael E.; Slaman, Theodore A. $\Sigma_2$-Collection and the Infinite Injury Priority Method. J. Symbolic Logic 53 (1988), no. 1, 212--221. https://projecteuclid.org/euclid.jsl/1183742576


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