Consider $(\mathbf{Q},<)$ as a structure in a language with one binary relation, which we interpret as the order. This is a universal, $\aleph_{0}$ categorical structure (see example of universal structure).

The theory of $(\mathbf{Q},<)$ has quantifier elimination, and so is o-minimal. Thus a type over the set $\mathbf{Q}$ is determined by the quantifier free formulas over $\mathbf{Q}$ which in turn are determined by the atomic formulas over $\mathbf{Q}$ An atomic formula in one variable over $B$ is of the form $x<b$ or $x>b$ or $x=b$ for some $b \in B$ Thus each 1-type over $\mathbf{Q}$ determines a Dedekind cut over $\mathbf{Q}$ and conversely a Dedekind cut determines a complete type over $\mathbf{Q}$ Let $D(p):=\{a \in \mathbf{Q}:x>a \in p\}$

Thus there are two classes of type over $\mathbf{Q}$

1. Ones where $D(p)$ is of the form $(- \infty,a)$ or $(-\infty,a]$ for some $a \in \mathbf{Q}$ It is clear that these are definable from the above discussion.
2. Ones where $D(p)$ has no supremum in $\mathbf{Q}$ These are clearly not definable by o-minimality of $\mathbf{Q}$