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On the T-Degrees of Partial Functions

Paolo Casalegno
The Journal of Symbolic Logic
Vol. 50, No. 3 (Sep., 1985), pp. 580-588
DOI: 10.2307/2274313
Stable URL: http://www.jstor.org/stable/2274313
Page Count: 9
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On the T-Degrees of Partial Functions
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Abstract

Let $\langle\mathscr{T},\leq\rangle$ be the usual structure of the degrees of unsolvability and $\langle\mathscr{D},\leq\rangle$ the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of $\langle\mathscr{D},\leq\rangle$: as a corollary, the first order theory of $\langle\mathscr{D},\leq\rangle$ is recursively isomorphic to that of $\langle\mathscr{T},\leq\rangle$. We also show that $\langle\mathscr{D},\leq\rangle$ and $\langle\mathscr{T},\leq\rangle$ are not elementarily equivalent.

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