Effective Theory - in Mathematics

In Mathematics

An effective theory is a formal theory whose set of axioms is recursively enumerable, that is, it is theoretically possible to write a computer program that, if allowed to run forever, would output the axioms of the theory one at a time and not output anything else.

Godel's first incompleteness theorem demonstrates that such a theory cannot at the same time be complete, consistent, and include elementary arithmetic. See also Proof sketch for Gödel's first incompleteness theorem#Hypotheses of the theory.