##  [Definable Set](/definable-set-1) 

 Definition

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

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

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

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

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

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

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

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

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.