By Joseph R. Shoenfield (Eds.)

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D. A set A i n X i s RE ir; H if t h e r e i s a space Y and a s e t B i n X X Y r e c u r s i v e i n H such t h a t x f o r all x. t o H. E A~+~Y( E B) We can t h e n r e l a t i v i z e a l l of t h e above r e s u l t s H i s a n I E Alg(X,N) such t h a t A = An index of A <. I f G is r e c u r s i v e i n H, t h e n any set RE i n G i s RE i n H (because of t h e corresponding r e s u l t with r e c u r s i v e i n p l a c e of RE). The jump o f a f u n c t i o n H i s t h e r e l a t i o n H* i n A l g ( N , N ) X N d e f i n e d by (1,r-D Then H* i s RE i n H .

It r e a d i l y f o l l o w s t h a t F ( T o t ) C T o t and F ( T o t C ) C F i n . A consequence o f t h i s r e s u l t i s t h a t e v e r y TT, reducible t o Finc. Now I E FinC* s o F i n C i s TT,. z2. plete set i s Vn3k(n < k & k E WI); n,; Thus F i n C i s complete s o F i n i s com- Now l e t Cof b e t h e set of I i n Alg(N,N) such t h a t WI We show t h a t e v e r y cofinite. Cof - v, s e t is i s reducible t o Tot, I n view of t h e above, i t s u f f i c e s t o f i n d a r e - Tot.

And t a k e A to b e t h e unique s e t such However, t h e t r e e s a r e t r e a t e d d i f f e r - A t s t e p s , we d e f i n e t r e e s T ently. For i n o t be t o t a l ) . < S i for i 5 ks (which need S ks, Tf+l w i l l b e a s u b t r e e of T:; S S 6, w i l l b e on Tk is and hence on a l l t h e T i . and W e w i l l always S have Tg = I d . A t t h e end of t h e c o n s t r u c t i o n , we show t h a t converges i n a s u i t a b l e s e n s e t o a t r e e Ti. T! Now we d e s c r i b e s t e p s .

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