The formal process of expanding a first-order language by adjoining new relation symbols (or predicate symbols) for sets and relations that are already definable in the original language, together with axioms that assert each new symbol is equivalent to the defining formula. The expansion makes definability explicit in the signature while remaining a conservative, definitional extension of theories that admit it.

Principle

Introduce a fresh symbol for each equivalence class of formulas defining the same relation and add axioms that identify the symbol with its defining formula; treat the expansion as a conservative definitional extension so that provability in the original language is preserved.

Demonstration

Given a language L and a family of L-formulas {φ(x) : φ defines a property of tuples}, form L' = L ∪ {R_φ : one new relation symbol per φ} and add axioms ∀x (R_φ(x) ↔ φ(x)). For a theory T in L that admits these identifications, T has a conservative extension T' in L' where each R_φ names the definable set of φ.

Misapplication

Adding arbitrary new predicates without giving axioms tying them to definable formulas or assuming that adding symbols for non-definable classes preserves conservativity. Confusing Morleyization with arbitrary nonconservative language enrichment that changes truth in models.

Consequence

Makes many model-theoretic arguments simpler by turning definability questions into syntactic ones about symbols; permits uniform use of new relation symbols in constructions (e.g., EM blueprints, indiscernibles) while controlling conservativity.

Reversal

Forgetting the added symbols (taking the reduct) returns to the original language; properties expressed solely via the new symbols may disappear on the reduct even though the expansion was definitional in the theory context.

Boundary

Applies to first-order definable sets and relations (or other explicitly given definable families); it does not convert genuinely higher-order, nonfirst-order, or nondefinable phenomena into named relations. Conservativity depends on choosing definitions that are provable in the base theory.

Semantic Tension

Competes with the notion of arbitrary conservative extension: Morleyization is a particular definitional conservative expansion that makes formulas into atomic symbols, whereas other expansions may be conservative for different reasons; tension arises when deciding whether to treat a relation as a primitive symbol or as a defined formula in proofs.

Synthesis

Morleyization is the controlled definitional expansion that replaces frequently used definable formulas by new predicate symbols plus equivalence axioms, thereby making definability explicit in the signature while preserving the original theory's content and enabling simpler syntactic and combinatorial manipulations.