Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 4 / f / iv / Solution

Consider the partial typeEach finite fragment is realized by a sufficiently large standard natural number, so it is consistent. It is omitted in the standard successor structure, since each of its elements equals one of the named iterates.

There is exactly one complete type extending this partial type. By classification of models of the successor theory and its quantifier-elimination property, a parameter-free one-variable formula is equivalent to a Boolean combination of finitely many equalities $x=s^n(0)$ and sentences whose truth is fixed by the theory. For a point on an extra integer chain every displayed equality is false, so all such points have the same complete type. Thusis the required omitted complete type. More formally take all formulas true at such a point; $p_{\infty }$ axiomatizes that type modulo the theory, and its deductive closure is complete.