# Finite and infinite

In mathematics, the meaning of the terms **finite** and **infinite** varies according to context.

Essentially, **finite** means (similar to common usage) having a size
which is *bounded* by a natural (or, equivalently, by a real) number.

while **infinite** means *unbounded* in size or, more precisely, exceeding all natural (or real) numbers in size.

(Often *bounded* and *unbounded* is used in the same sense.)

But "size" may mean length, area, or the result of any other measurement,
and thus the precise meaning of "finite" varies accordingly,
but is often not explicitly given.
Examples are:

finite interval, finite value, finite integral,
finite degree, finite dimension, finitely often, etc.

A special case of size is cardinality,
i.e., size with respect to the number of elements:

**finite** sets have finitely many elements, i.e., 0 or 1 or 2 or 3 ... elements,

**infinite** sets have more (i.e., at least an unlimited sequence of) elements.

Thus the interval of real numbers between 0 and 1 is
a *finite interval* and a *bounded set* because its *length* is bounded,
but it is an *infinite set* because it contains infinitely many numbers.