,...} is unbounded
literally if it is not bounded
: for any N we can find some j
for which |xj
|>N. Note that all the values xi
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.