A definable set in a structure M is a subset of M^n that equals the solution set of a first-order formula φ(x) possibly with parameters from M; that is, D = { a ∈ M^n : M ⊨ φ(a) } for some φ and parameter tuple from M.

Principle

Definability means there is a uniform first-order description (a formula) that picks out exactly the members of the set in the given structure, possibly using parameters to name fixed elements.

Demonstration

In a field, the set of zeros of a polynomial f(x) with coefficients in the field is definable by the atomic formula f(x)=0; in a group, the centralizer of an element g is definable by the formula xg=gx.

Misapplication

Confusing definable sets with algebraic or topological sets: a set definable by a first-order formula might not be an algebraic variety, and conversely algebraic sets in some languages require parameters or expansion to be definable.

Consequence

Definable sets are closed under Boolean combinations and under projection (existential quantification), so they form the basic measurable pieces for model-theoretic analysis of a structure and determine definable functions and relations.

Reversal

The inverse notion is type-definable or invariant sets: an intersection of possibly infinitely many definable sets (type-definable) is generally not definable by a single first-order formula, showing a weakening of uniform definability.

Boundary

Definability depends on the chosen language and allowed parameters; some subsets are definable only after naming constants or expanding the language, and definability excludes infinitary or higher-order descriptions unless explicitly allowed.

Semantic Tension

‘Definable’ competes with ‘interpretable’ and ‘type-definable’: interpreted sets may require coding across sorts, while type-definable sets are intersections of definable sets; the distinctions affect transfer of properties between structures.

Synthesis

A definable set is a subset of a structure singled out by a single first-order formula (with possible parameters); it provides the canonical building block for studying the geometry and combinatorics of models in first-order terms.