Abstract

We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an e#ective version of coding lemma, we show that the first order theory of the enumeration degrees of the # 0 2 sets is not decidable. 1 Introduction Definition 1.1 The following terms specify the variations on relative enumerability. . An recursive enumeration procedure U is a recursively enumerable collection of pairs #a, b# in which a and b are finite subsets of N. . U(A) is equal to B if B = {n : (#a)(#b)[a # A and #a, b# # U and n # b]} . . Say that B is enumeration reducible to A (A # e B) if there is a recursive enumeration procedure U such that U(A) = B. Similarly, A is enumeration equivalent to B (A # e B) if A # e B and B # e A. Definition 1.2 The enumeration degrees are defined as follows. . The enumeration degree of ...

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