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Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees

Harold T. Hodes
The Journal of Symbolic Logic
Vol. 45, No. 2 (Jun., 1980), pp. 204-220
DOI: 10.2307/2273183
Stable URL: http://www.jstor.org/stable/2273183
Page Count: 17
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Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees
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Abstract

Where $\underline{a}$ is a Turing degree and ξ is an ordinal $< (\aleph_1)^{L^\underline{a}}$, the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$, is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

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