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Effective theory has two distinct meanings.

In science

An effective theory is a scientific theory which proposes to describe a certain set of observations, but explicitly without the claim or implication that the mechanism employed in the theory has a direct counterpart in the actual causes of the observed phenomena to which the theory is fitted. I.e. the theory proposes to model a certain effect, without proposing to adequately model any of the causes which contribute to the effect.

Thus, an effective field theory is a theory which describes phenomena in solid-state physics, notably the BCS theory of superconduction, which treats vibrations of the solid-state lattice as a "field" (i.e. without claiming that there is "really" a field), with its own field quanta, called phonons. Such "effective particles" derived from effective fields are also known as quasiparticles.

In a certain sense, quantum field theory, and any other currently known physical theory, could be described as "effective", as in being the "low energy limit" of an as-yet unknown "Theory of Everything".[1]
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.
References

^ c.f. e.g. Ion-Olimpiu Stamatescu, Erhard Seiler, Approaches to fundamental physics: an assessment of current theoretical ideas, Lecture notes in physics, vol. 721, Springer, 2007, ISBN 978-3-540-71115-5, p. 47