nonrealizable

Nonrealizable is an adjective used in several mathematical and logical contexts to describe objects, propositions, or structures for which there is no constructive witness or representation within a given framework. In logic and constructive mathematics, realizability assigns to each statement a computational witness, usually in the form of a program or term. A proposition is realizable if such a witness exists; it is nonrealizable when no witness can be found inside the chosen realizability semantics. This distinction highlights differences between classical provability and constructive meaning, since some classically provable statements may be nonrealizable in a constructive setting.

In realizability theory, a statement’s nonrealizability can indicate limits of certain computational interpretations. For example, while the law of excluded middle is classically valid, it often fails to be realizable in intuitionistic or type-theoretic realizability models, reflecting the constructive requirement of explicit evidence.

In combinatorics and matroid theory, nonrealizable (or non-representable) refers to a matroid that cannot be realized as a linear matroid over any field. Such objects are studied to understand the boundaries between abstract combinatorial properties and linear representability. Nonrealizable matroids have connections to geometry, coding theory, and advanced matroid theory, and they illustrate that some abstract configurations lack a linear realization.

The term also appears in other domains to mark the absence of a concrete, implementable, or measurable realization within a specified framework. See also realizability, intuitionistic logic, and nonrepresentable structures.