 By Melvin Fitting

Hardbound.

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Extra info for Fundamentals of Generalized Recursion Theory

Example text

9. Let d be infinite. e. over '? d in 9,,. . e. ,gk. 8. 14. R-definability There have been many attempts to define a notion of recursion theory suitable for arbitrary structures, and they have led in a variety of directions. But a surprising number of them have turned out to be in the same family, in that they give essentially equivalent results. This may be counted an argument for naturalness, if not for philosophical correctness. The best known member of this family is search computability, from Moschovakis in .

DEFINITION. E:' is the equality operator of order (n, n ) on the ith and jth coordinates. , x , ) E 9 ' ~ x x , = x , } . 3. Each EZj is in A. If i = j, E:, is I" which is in A. Next we show E E A. Then . * A 3 A z @is constant, of order (n, n ) , and value the operator A d x d x .. 4. There are "place-duplicating" operators in A. That is, for each n, i, j, there is an operator @ E A of order (n, n + 1 ) that inserts a duplicate of the ith place at position j . , ~ x , + ~ ) € 9). We show the special case where i = 2, j = 1; the general case then follows using transpositions.

E. over H. Every relation existential inductive on d in . e. over 3 . Suppose 3 is inductive on d in 3,, . . ,%k. Then there is a fixed point 9,and there are a l , .. , a, E d , with x E3 ( a , x ) E 9. e. over H. Say it is represented by P in the elementary formal system with axioms E. Add to E the axiom Pa, x + Rx. e. I. The second part is similar. DEFINITION. Let E be a set of axioms for an (w) elementary formal system. We call E simple if, besides the predicate symbols R,, . . ,Rk,there is only one other predicate symbol in E.