Theory of (Z,+) has uncountably many 1-types

Question

I'm working on some exercises in model theory, but on this one I don't know how to start. Please help to solve this.

Prove that $\text{Th}(\mathbb{Z},+)$, the theory of the structure $(\mathbb{Z},+)$, has uncountably many $1$-types.

Answer 1

A hint is: what sorts of sets can you define in this theory? Think about $\phi(x)=\exists y . x+py=c$.

A direct answer is given by noticing that, for any $c\in \prod_p \mathbb Z/p\mathbb Z$ where $p$ ranges over the primes, every finite set of formulas of the form $\exists y . x+py=c_p$ is realisable. By Zorn's lemma we can pick a complete 1-type $t_c$ extending the set $\{\exists y . x+py=c_p \mid p\text{ prime}\}$, and if $c\neq c'$ then $t_c\neq t_{c'}$. If you want to remove the use of the axiom of choice, you'll have to define the complete 1-types directly by being more careful (I suggest looking up profinite integers).