In mathematics, a mathematical object X of some type T is definable, if there exists some predicate P(x) which is expressible using a finite string of mathematical symbols drawn from a finite language, such that P(X) is true and P(Y) is false for all Y of type T such that X <> Y.

All computable objects are definable, but not all definable objects are computable.

Thus, we have definable numbers, definable sets , definable sequences , definable functions , etc.

Classical mathematics permits (and requires) the existence of undefinable objects. Some people find this philosophically disquietening, questioning how an object can be said to exist if no mathematical statement can be used to uniquely identify it.

As a result, a few mathematicians have developed systems of mathematics that do not involve undefinable objects.

Notably, in a system of mathematics in which everything is definable, all sets are countable.