example of definable type

Consider as a structure in a language with one binary relation, which we interpret as the order.This is a universal, ${\mathrm{\aleph }}_0$-categorical structure (see example of universal structure).

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

Thus there are two classes of type over $𝐐$.

1. Ones where $D(p)$ is of the form $(-\mathrm{\infty },a)$ or $(-\mathrm{\infty },a]$ for some $a\in 𝐐$. It is clear that these are definable from the above discussion.

2. Ones where $D(p)$ has no supremum in $𝐐$. These are clearly not definable by o-minimality of $𝐐$.