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**1 - 1**of**1**### THE THEORY OF THE α DEGREES IS UNDECIDABLE

"... Abstract. Let α be an arbitrary Σ1-admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are α-degrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of α-degrees is undecidabl ..."

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Abstract. Let α be an arbitrary Σ1-admissible ordinal. For each n, there is a formula ϕn(⃗x, ⃗y) such that for any relation R on a finite set F with n elements, there are α-degrees ⃗p such that the relation defined by ϕn(⃗x, ⃗p) is isomorphic to R. Consequently, the theory of α-degrees is undecidable. 1.