Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / B / 7 / iv

Take the parameter-free complete typeof the nonstandard prime constructed in part (i). It is first-order consistent and complete because it consists of exactly the logical formulas true of an actual element in a first-order model of $T$. It contains $x\ne \bar{a}$ for every integer $a$, and consequently no element of $\mathcal{Z}$ realizes it. ThusIt is necessarily a nonprincipal complete type. If a first-order consistent logical formula isolated it, completeness of $T$ would force the existential closure of that logical formula to hold in every first-order model of $T$, including $\mathcal{Z}$; a witness there would realize the partial type, a contradiction. This connects the explicit omission with the hypothesis of the omitting types theorem.