(logic, mathematics):
A sequence {x1,x2,...} 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 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.

Un*bound"ed, a.

Having no bound or limit; as, unbounded space; an, unbounded ambition.

Addison. -- Un*bound"ed*ly, adv. -- Un*bound"ed*ness, n.

 

© Webster 1913.

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