Beth definability

In mathematical logic, the Beth definability theorem is a fundamental result in logic that states a property implicitly defined by a first-order theory has an explicit definition within that theory.This means that if a new symbol or property is uniquely determined across all models of a theory, there must be a formula using only the existing symbols of the theory that defines it. The theorem establishes an equivalence between implicit and explicit definability in classical first-order logic, linking its syntax and semantics.

Statement

For first-order logic, the theorem states that, given a theory T in the language L' ⊇ L and a formula φ in L', then the following are equivalent:

for any two models A and B of T such that A|''L = B''|L, it is the case that A ⊨ φ if and only if B ⊨ φ ;φ is equivalent modulo T to a formula ψ in L.

Less formally: a property is implicitly definable in a theory in language L only if that property is explicitly definable in that theory.
Clearly the converse holds as well, so that we have an equivalence between implicit and explicit definability. That is, a "property" is explicitly definable with respect to a theory if and only if it is implicitly definable.
The theorem does not hold if the condition is restricted to finite models. We may have A ⊨ φ if and only if B ⊨ φ for all pairs A,''B of finite models without there being any L''-formula ψ equivalent to φ modulo T.
The result was first proven by Evert Willem Beth in a paper published in 1953.