For each prime $p$, we have the algebraically closed field $\bar{\mathbb F}_p$ with the Frobenius automorphism.

Given any first-order statement with no free variables using the symbols $0,1, +, \times, -, /, \sigma(),=$, we can interpret it in $\bar{\mathbb F}_p$, interpreting the field operations to mean themselves and $\sigma$ to mean Frobenius.

For each prime, it is either true of false. This gives us a set of primes.

What sets of primes can be described this way?

It is easy to pick out the primes whose Frobenius elements have a certain conjugacy class in in a Galois extension of $\mathbb Q$, and to pick out finite sets of primes.

Are there any sets of this type not generated by conjugacy classes in Galois groups and finite sets under the logical operations?