Cut types over the named rational order ($p_r(x)=\{q<x:q<r\}\cup \{x<q:q>r\}$)

For each irrational real cut $r$, the displayed comparisons with named rationals are finitely satisfiable. Quantifier elimination for dense linear orders makes them determine a complete type, omitted by the named rational structure itself. Different cuts give different types, so there are continuum many. An omega-saturated model must realize all of them and cannot be countable, although the named rational structure is countable, atomic and prime.