On relative coimmunity

Abstract

are sets of numbers such that A and -A is immune, we say that is coimmune in . (In case A -Wj, -W k , for some j and k, we say instead that is simple in .) Similarly, if = and -A is hyperhyperimmune, we say that A is cohyperhyperimmune in 2 1 , or that A is hyperhy per simple in 9 in case both z/ and are r.e. (For definition and discussion of the notion of hyperhyperimmunity, the reader may consult [9] or [10]; the existence of hyperhypersimple sets is known from [5].) LEMMA 1. There exists a set of numbers, a, such that both a and its complement, a, are hyperhyperimmune.