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

Question

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?

Answer 1

Rather than giving an explicit answer, I'm going to give a hint. I saw this problem when I was in grad school, and I assume many other people did too. The secret to these problems is to have a huge library of mathematical results to draw on. Then you make up your answer to exploit some theorem that you already know. In this case, one way to start is to make a formula which forces the model to resemble an initial segment $\{1, \ldots, n\}$ of the natural numbers with relations for the restrictions of the graphs of the addition and multiplication functions to triples from that subset.

Answer 2

This exists due to very general results, namely that the set of primes is rudimentary. See this excellent survey on spectra: Durand et al. Fifty Years of the Spectrum Problem: Survey and New Results. The same holds true for all known "natural" number theoretic functions. Indeed, the authors remark in section 4.2 that "we are not aware of a natural number-theoretic function that is provably not rudimentary".

Answer 3

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 $A'\wedge\neg B$.