Double Recursion Theorems

26 Chapter I. Recursive Enumerability and Recursivityfrom R(XI, ..., x n ) if there are functionseach of which is either a projection function or a constant function,such that for all xi,...,Xk'For example, suppose S = \x\x^xzx^: $(2:3,7, £2)- ThenUnions and Intersections. For any two relations R\(XI, ... ,a; n )and R?(XI, • • • 5 x n)i by R\ U R% (the union of R\ and R^), we meanand by the intersection R\ n R? of Ri,R%, we meanWe know from Th. 4, Ch. 0 that if R\ and R% are both r.e., then soare the relations R\ U R^ and R\ n R%.Quantifications. For any relation R(XI ,..., a; n , y) by its existentialquantification, we meanand by its universal quantification, we meanWe know from Chapter 0 that the existential quantification of anr.e. relation is r.e. (because existential quantifications of S-relationsare S, and S-relations are the same as EI-relations). The universalquantification of an r.e. relation is in general not r.e. (as we will see).Finite Quantifications. By the finite existential quantification ofa relation R(x\,..., x n , y, z), we meanBy the finite universal quantification of J?, we meanBy Th. 4, Ch. 0, if R is r.e., then the finite existential quantificationof R and the finite universal quantification of R are both r.e.