A recursively enumerable set 
is creative if its complement is productive.
Creative sets are not recursive. The property of creativeness coincides with completeness.
Namely, set 
is creative iff if it is many-one
complete.
Elementary arithmetic formulas are built up from 0, 1, 2, ..., 
, 
,
, variables,
connectives, and quantifiers.
The set of all true arithmetic formulas is productive. Informally speaking, this
means that no axiomatization of arithmetic can capture all true formulas and nothing
else. For example, consider Peano arithmetic.
Under the assumption that no false arithmetic formulas are provable in this theory,
provable Peano arithmetic formulas form a creative
set.
