Nonstandard element type ($p_{\infty }(x)=\{x>\overline{n}:n\in \mathbb{N}\}$)

In a model of Peano arithmetic, an element realizes this partial type exactly when it is not a standard numeral. A model omitting it is isomorphic to the standard natural-number structure. For a complete consistent extension $T$ of arithmetic, the type is nonprincipal exactly when $T$ is the Theory of true arithmetic: apply the omitting types theorem in one direction and use an actual standard witness to refute any proposed isolating formula in the other. For incomplete theories, a principal partial type can still be omitted by some models.