A first order sentence such that the finite Spectrum of that sentence is the prime numbers 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?

The set of prime numbers and its complement are spectra. Consider the language $\mathcal{L}$ (which contains only predicates and equality) and the $\mathcal{L}$-sentence $A'$ I define in this answer (you will also find how the predicates I use next are defined). Now, let $B$ be the $\mathcal{L}$-sentence saying that a prime number is the largest element in our domain:
$$\exists z\,((\neg\exists w\, Rzw)\wedge(\neg\exists x\,\exists y\,(Rxz\wedge Ryz\wedge Qxyz))\wedge(\neg Oz)).$$
The set of prime numbers is the spectrum of $A'\wedge B$, and its complement, the set of composite numbers, is the spectrum of $A'\wedge\neg B$.
