Past exam of the model theory course of the University of Oxford / 2011 / 1 / c / iii / Solution

For the additive first-order structure, letIt defines $n\mathbb{Z}$. These sets are pairwise distinct: if $m>n>0$, the integer $n$ belongs to $n\mathbb{Z}$ but not to $m\mathbb{Z}$. Thus there are infinitely many inequivalent one-variable first-order formulas modulo the complete theory of the additive integers. The Ryll-Nardzewski theorem implies that first-order theory is not aleph-zero-categorical.

For the multiplicative first-order structure define divisibility by $d(y,x):=\exists z(yz=x)$. For $k\ge 1$, the first-order formulaasserts the existence of at least $2k$ distinct integer divisors. The integer $2^{k-1}$ has exactly $2k$ such divisors, namely $\pm 2^j$ for $0\le j\le k-1$, so it satisfies ${\psi }_k$ but not any ${\psi }_{\ell }$ with $\ell >k$. Again there are infinitely many inequivalent one-variable first-order formulas. The Ryll-Nardzewski theorem therefore implies the first-order theory of the multiplicative integers is not aleph-zero-categorical.