By Klaus Ambos-Spies (auth.), Egon Börger, Walter Oberschelp, Michael M. Richter, Brigitta Schinzel, Wolfgang Thomas (eds.)

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Additional resources for Computation and Proof Theory: Proceedings of the Logic Colloquium held in Aachen, July 18–23, 1983 Part II

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The second C as oracle, via a C-correct compu- implies B [ x u = B [ x , m m m So R acts only finitely often. It remains to show that R is met. For a contran n assume that A = {n} C, and choose s 3 > s 2 s u c h t h a t Rn d o e s n ' t act after diction stage s 3. Note that there are infinitely many TFn-stages. , x and since otherwise a new follower. thus, as pointed at the So Rn has certain followers least By definition out above, stage s+l > s 3 where of s3, the followers completely confirmed at Xl, sta~e m s 3.

2, choose r°e. sets A. , Bn ( i # j + (i < n,~ ~ F) degB i n degBj = O )). such that with n-] in place of n. The embedding f : U -~ R(<_aa) is defined by n f (~) = O and, for ~ a U m - {~} f(~) = deg(A i where • ... ~ A. 13). (4. 16) f(~uB) = f(~) u f(B). e. 16) c U . It remains to show that f is onem ~,B ~ % , ~ # B ÷ f(~) # f(B), and that f preserves infima. e. degree c, if c < f(~) and c < f(B) then c < f(c~B). ]6) fix ~,~ s U such that ~ ~ ~. g. B ~ ~, say m is not in ~ but in B. Then degAi,{p } _< f(~).

Mp-rank(~). 5 requires 1emma. (n > 2) be the closure under finite unions of the set n i < n, the following k c ~} u {@}. Then ( i) U is a sublattice of F with least element ~. n ( ii) mp-rank(Un) (iii) = n. ,i but somewhat k < n and tedious. |3) where ie ~ = i ~ u . n+i Proof of Corollary of a is greater O < i < . 5. : k E ~} For the nontrivial to show U o n ... n ~k' • direction than or equal to n. 5. 3, we may assume that n >__ 2. So, =o=> ~(

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