By John T. Baldwin (auth.), Manuel Lerman, James H. Schmerl, Robert I. Soare (eds.)

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Qk are q u a d r a t i c yields a nontrivial Qk(C)a k forms. If d > 2k then the C h e v a l l e y theorem [ 4 ] s i m u l t a n e o u s z e r o ~ o f the Qi i n Fp, and then Zcix i E V, a contradiction. LEMMA 8 - If k = 1 then J is i s o m o r p h i c t o A2, Cp o r D p ( t ) . Proof : If dim J/V = I i t i s easy t o see t h a t J is i s o m o r p h i c w i t h A2 ( s i n c e e v e r y e l e m e n t o f V has a square r o o t , other characteristics are r u l e d o u t ) . Lemma 7 we may now suppose t h a t dim J/V = 2.

C o n t i n u e d ! f(x). e, Note that this extends What do we know? e, the last place Let's As Certainly for some changes T h e o r e m 4. e for some we can spot recursively class. it) in 0' Abbreviate identity- degree. e. argument for some ~= D(< 0') then is undecidable. e. e, is the construction set, by allowing y ! x. , [3]). } T <7) : A Theorem 5: A Let arguof <7) is undecidable We can also ask w h e t h e r is the same as <)) sets. e. e. $ be a class of total recursive we can recursively in by permitting <7).

In addition we need a new trick here not found in the proof of the minimal pair. its A-restraint) Namely, a~ is allowed only at a stage close that gap (thereby reimposing (providing is an e-expansion t for all to allow enumeration in However A-restraint) N~-gap a~ (and drop is allowed to at a stage t ~ Sk stage for the corresponding allows us to impose a sufficiently lim inf s r(n,s) < ~ to open an s c S k. small amount of restraint e). This so that n, and yet close the gap often enough Ce and meet Lemma 3 below.