 Numeralwise enumeration in arithmetic

A two-variable formula enumerates a set $A$ in a theory $S$ when membership supplies a standard numeral stage whose instance is provable, while nonmembership makes every standard-stage instance refutable. The notion contains these proof obligations, not merely a syntactic complexity restriction on the formula. With bounded completeness, a bounded formula whose existential closure defines a computably enumerable set provides such an enumerator.^[ref]