Editing Infinite divisibility (section)

==In order theory==

To say that the [[field (mathematics)|field]] of [[rational number]]s is infinitely divisible (i.e. order theoretically [[Dense set|dense]]) means that between any two rational numbers there is another rational number.  By contrast, the [[ring (mathematics)|ring]] of [[integer]]s is not infinitely divisible.

Infinite divisibility does not imply gaplessness: the rationals do not enjoy the [[supremum|least upper bound property]].  That means that if one were to [[partition of a set|partition]] the rationals into two non-empty sets ''A'' and ''B'' where ''A'' contains all rationals less than some irrational number (''[[Pi|π]]'', say) and ''B'' all rationals greater than it, then ''A'' has no largest member and ''B'' has no smallest member.  The field of [[real number]]s, by contrast, is both infinitely divisible and gapless.  Any [[total order|linearly ordered set]] that is infinitely divisible and gapless, and has more than one member, is [[uncountable set|uncountably infinite]].  For a proof, see [[Cantor's first uncountability proof]].  Infinite divisibility alone implies infiniteness but not uncountability, as the rational numbers exemplify.