well-defined

Suppose you label people by their phone number. The fraction 1/2 and the fraction 2/4 are the same person wearing two name tags. A rule on fractions is well-defined only if it gives the same answer whether you read the first tag or the second — otherwise the rule depends on the tag, not the person, and is incoherent.

A definition is well-defined when it specifies a single, unambiguous object that does not depend on the arbitrary choices made to describe its input — for example, the representative of an equivalence class, the labelling of elements, or the order of a construction. Whenever an object can be named in more than one way, you must check the proposed definition assigns the same value to every name.

This is not a triviality but a genuine obligation of proof. The limit of a sequence is well-defined because one can prove it is unique; addition of real numbers built from Cauchy sequences is well-defined because the sum's limit does not depend on which Cauchy sequences represent the two reals. Skipping the check is a classic source of error: a ‘function’ that hands back different outputs for the same input is not a function at all.