TITLE: A first order sentence such that the finite Spectrum of that sentence is the prime numbers
QUESTION [8 upvotes]: The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ . What I am asking for is a finitely axiomatized $T$ such that $Fs(T)$ is the set of prime numbers. 
In other words in what specific language $L$, and what specific $L$-sentence $\phi$ has the property that $Fs(\{\phi\})$ is the set of prime numbers?

REPLY [3 votes]: Consider finite fields equipped with a total order such that for every $x$, if it has an immediate successor, then that successor is $x+1$.