Double Recursion Theorems

8 Chapter 0. PrerequisitesOne also applies the notion of separability to relations of naturalnumbers. We say that F(VI,. .. ,v n ) weakly separates R\(XI,. .. ,ar n )from RZ(XI,. .. ,x n ) if H(a\,... ,a n ) is provable for every n-tuple(ai,..., a n ) in RI and for none in R%. F is said to strongly separateRI from RI if F is provable for every ra-tuple in jRi and refutable forevery n-tuple in R^.§3.1. Definability and Complete Representability. Weregard sets (of natural numbers) as special cases of relations (setsare relations of one argument). We say that a formula F(v\,',..., v n )defines a relation R(XI, ..., x n ) in S if F strongly separates R fromits complement R—in other words, if F(ai,... a n ) is provable for everyra-tuple 01,...,a n such that R(ai,... ,a n ) holds and is refutablein S for every n-tuple (ai,..., a n ) such that ~ jR(ai,..., a n ) holds.We say that F completely represents R in S if F represents R, andits negation ~ F represents the complement R. For a consistentsystem 5, definability and complete representability are equivalent(as the reader can easily verify).Let us note that if a formula F(v-i,V2) defines a relation R(x,y)in