The Π₃-Theory of the [image] -Enumeration Degrees Is Undecidable

Journal of Symbolic Logic 71 (4):1284 - 1302 (2006)
We show that in the language of {≤}, the Π₃-fragment of the first order theory of the $\Sigma _{2}^{0}$-enumeration degrees is undecidable. We then extend this result to show that the Π₃-theory of any substructure of the enumeration degrees which contains the $\Delta _{2}^{0}$-degrees is undecidable
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DOI 10.2178/jsl/1164060455
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John Case (1971). Enumeration Reducibility and Partial Degrees. Annals of Mathematical Logic 2 (4):419-439.

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