By Joseph R. Shoenfield (Eds.)

ISBN-10: 0080871135

ISBN-13: 9780080871134

ISBN-10: 072042061X

ISBN-13: 9780720420616

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We u s e T t o d e s i g n a t e t r e e s . A string i s 2 a t r e e T i f i t b e l o n g s t o t h e r a n g e of T . A set A i n N is 0" T if w C A f o r i n f i n i t e l y many on T . 4( We s a y t h a t T I i s a s u b t r e e o f T i f e v e r y s t r i n g on T I i s on T . T h i s i m p l i e s t h a t e v e r y set on T I i s on T . We s a y t h a t [I]& and [I]' O( and 8 I-split 7 a r e incompatible. 49 if 7 C o< , &? , and o< and 6 y c This implies t h a t MINIMAL DEGREES 50 -y ; for if 6 were a common e x t e n s i o n of o< and G i B [ I ] ' would be a common e x t e n s i o n o f [ I ] and [ I ] .

D. 10. The Jump O p e r a t i o n We a r e g o i n g t o d e t e r m i n e t h e range of t h e jump o p e r a t i o n . By (12) of range i s > 0' i s 46, 0' 2 0'. 5 a ' for a l l a ; s o e v e r y d e g r e e i n t h i s We now show t h a t , c o n v e r s e l y , e v e r y d e g r e e t h e jump of some d e g r e e . Theorem ( F r i e d b e r g ) . I f 0' such t h a t b 1 = b u 0' 5 a , then t h e r e i s a degree b a. = While t h i s theorem g i v e s a n i c e p r o p e r t y of t h e jump o p e r a t i o n , i t can be used t o show t h a t t h i s o p e r a t i o n h a s p r o p e r F o r example, i t i s n o t one-one.

We c a n a l s o compute t h e n of Now Case 1 h o l d s i f f (1) S i n c e {Rsf & ~ U ( C O ~ ~ ( U , F ~ )n E w,O). rr) i s a r e c u r s i v e r e l a t i o n of a, TT, (1) i s an RE r e l a t i o n o f FS, I, n ( b y t h e r u l e s of t h e l a s t s e c t i o n ) . T h i s r e l a t i o n i s t h e n of d e g r e e 5 0 ’ . Given an o r a c l e for t h i s r e l a t i o n , we can d e c i d e which c a s e h o l d s . If w e a r e i n Case 2, I f we a r e i n t h e r e i s no problem. Now t h e Case 1, t h e o n l y remaining problem I s t o f i n d (3.

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Degrees of Unsolvability by Joseph R. Shoenfield (Eds.)


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