Journal of Symbolic Logic

Uniform Enumeration Operations

A. H. Lachlan

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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 $\mathbf{d}$. 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 $\mathbf{d}$. In addition, it is proved that the only other uniform enumeration operations such that $\mathbf{d} \leq W (\mathbf{d})$ for all $\mathbf{d}$ are those which equal the identity operation above some fixed degrees.

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J. Symbolic Logic, Volume 40, Issue 3 (1975), 401-409.

First available in Project Euclid: 6 July 2007

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Lachlan, A. H. Uniform Enumeration Operations. J. Symbolic Logic 40 (1975), no. 3, 401--409.

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