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E. set D so that B = A © D has the desired properties, namely D < T A, and B does not support C. We ensure the condition D < T A by a permitting argument. To satisfy the last property we meet the following requirements for all e, Se : $ e total & (M, —► $ e is not an extension of Z/, where { $ e } e G a ; is an enumeration of all partial- recursive functions. For each e G w w e set aside the set of numbers Re = {2(e, x) + 1 : x 6 CJ} from which to draw witnesses to satisfy Se.

The reader will notice that many of these had originated in ordinal recursion theory developed since the early 1960's. The fact that there is a tight link between recursion theory on ordinals and that on weak fragments of P is a fortuitous coinci­ dence probably not anticipated by the founders of these two subjects. We believe that there is a two-way interaction between them, and expect to see 48 C H O N G AND Y A N G more applications going in either direction. We begin with some definitions and terminologies.

P. e. comeager. Proof. It suffices to define an effective system of partial extension functions / such that for every set A £ BI(C) there is an index e such that fe is dense along A and fe(A\n) % A whenever fe(A\n) is defined. Given a recursive set U such that C = {Ue : e > 0} such a system / is obtained by letting f2e+i(x) — x~i if \x\ £ Ue and / 2 e + i ( x ) | otherwise (e > 0, i < 1). e. nowhere dense via / 2 e or / 2 e + i , respectively. e. meager but not effectively meager. e. meager but not effectively meager.