apartness

Apartness is a binary relation used in constructive mathematics and related fields to express a positive, verifiable difference between two objects. It provides a constructive witness that two elements are distinct, in contrast to the classical notion of inequality which is often the negation of equality.

Formally, an apartness relation # on a set X is typically required to be irreflexive (no element is apart from itself), symmetric (if x # y then y # x), and cotransitive (if x # y then x # z or z # y for all z). These axioms encode a robust sense of disjointness that supports constructive reasoning about distinctness. In many frameworks, equality is handled in relation to apartness, with some systems taking x = y to mean not (x # y); in others, equality remains a separate primitive and apartness provides a positive means to witness differences.

A common example is the apartness on constructive real numbers. Two reals x and y are apart, written x # y, if there is a constructive positive separation between them, often realized by a witness such as a rational bound r > 0 with |x − y| > r. A typical constructive formulation uses a witness like |x − y| > 2^{-n} for some natural number n. This makes apartness a stronger, more informative notion than mere inequality in classical logic, since it supplies explicit evidence of the difference.

Applications of apartness appear in constructive analysis, topology (as apartness spaces), and algebra (where it helps manage constructive notions of separation and discreteness). It also features in type theory and computer-assisted proofs, where providing a constructive witness of distinctness is essential.

See also: constructive mathematics, equality, inequality, apartness space.