The set of identities provable in primitive recursive arithmetic without induction (PRE), with or without standard quantifier free successor axioms, is recursive. A finite number of identities can be added to PRE such that the set of identities provable become complete r.e. (with or without successor axioms). If the successor axiom y Φ 0 -» 3x(S(x) = y) is added to PRE, then the set of identities provable become complete r.e. (with or without l=£0). If PRE is augmented by definition by cases,

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... en the set of identities provable become complete r.e. (with or without successor axioms). Equivalents of primitive recursive arithmetic (PRA) are given, involving the rule of induction restricted to identities. We begin with the definition of the %-ary primitive recursion indices, 0 < n. These are given by (i) 0, 1 are 1-ary p.r. indices, (ii) (2, n, m), 1 5£ m ^ n, is an n-ary p.r. index, (iii) (3, x, y lf , y k ) is an %-ary p.r. index if y l9 •••, y k are %-ary p.r. indices and x is a A -ary p.r. index, 1 ^ n, k. (iv) (4, x, y) is a 1-ary p.r. index if y is