forall p,q,r in Ps, if p>q>r => there exists a,b in [0,1] s.t. ap+(1-a)r>q>bp+(1-b)r (archimedean axiom) This axiom says there is nothing incomparably good or incomparably bad.

This axiom is satisfied on the real line.

Independence often fails in real life. People often place death as incomparably bad.