On The Filter Of Computably Enumerable Supersets Of An R-Maximal Set

. We study the filter L (A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in L (A) is still \Sigma 0 3 -complete. This implies that for this A, there is no uniformly computably enumerable "tower" of sets exhausting exactly the coinfinite sets in L (A). 1. The theorem The computably enumerable (or recursively enumerable) sets form a countable sublattice (denoted by E in the following) of the power set P(!) of the set of natural numbers. The operations of union and intersection are effective on E (i.e., effective in the indices of the computably enumerable sets). The complemented elements of E are exactly the computable sets. The finite sets in E are definable as those elements only bounding complemented elements; thus studying E is closely related to studying E , the quotient of E modulo the ideal of finite sets. (From now on, the superscript will always denote that we are wor..