(

logic,

mathematics):

A

*sequence* {

`x`_{1},

`x`_{2},...} is

*unbounded* literally if it is not

bounded: for any N we can find some

`j` for which |

`x`_{j}|>N. Note that all the values

`x`_{i} are

*finite*!

By analogy, *unbounded* means "(finite, but) with no *a priori* bound". For instance, the set of English utterances is unbounded: for physical reasons only finitely many utterances can ever be produced, but there's no *a priori* bound on them. Similarly, Euclid's geometry takes place on an unbounded plane: *every* construction takes place in a finite area of the plane, but this area is *not* bounded. Indeed, Euclid recognised this; his lines aren't infinite, but rather "indefinitely extensible"!

Other terms for the same idea include potentially infinite and s_alanet's theoretically infinite, but "unbounded" is the preferred modern terminology.