The Π₃-Theory of the $\Sigma _{2}^{0}$ -Enumeration Degrees Is Undecidable

Journal of Symbolic Logic 71 (4):1284 - 1302 (2006)
Abstract
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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A. H. Lachlan (1975). Uniform Enumeration Operations. Journal of Symbolic Logic 40 (3):401-409.
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