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≠0 → gx(S(x) = y) is added to PRE, then the set of identities
provable become complete r.e. (with or without 1≠0). If PRE is augmented
by definition by cases, then 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.
