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Uniform Enumeration Operations

A. H. Lachlan
The Journal of Symbolic Logic
Vol. 40, No. 3 (Sep., 1975), pp. 401-409
DOI: 10.2307/2272164
Stable URL: http://www.jstor.org/stable/2272164
Page Count: 9
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Uniform Enumeration Operations
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Abstract

Sacks [2] has asked whether there exists a uniform solution to Post's problem, i.e. an enumeration operation W such that $\mathbf{d} < W(\mathbf{d}) < \mathbf{d}'$ for every degree d. It is shown here that if such an operation W exists it cannot itself in a particular technical sense be uniform. In fact, the jump operation is characterized amongst such uniform enumeration operations by the condition: $\mathbf{d} < W(\mathbf{d})$ for all d. In addition, it is proved that the only other uniform enumeration operations such that d ≤ W (d) for all d are those which equal the identity operation above some fixed degrees.

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