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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?

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In what language? – Chris Eagle Mar 20 '11 at 23:10
Are $0$ or $1$ prime by your definition? – Bill Dubuque Mar 20 '11 at 23:25
@Bill: That doesn't matter. There's a formula $\phi_n$ (in the empty language) such that $\mathcal{M} \models \phi_n$ iff $| \mathcal{M} | = n$. By disjoining such formulae, or conjoining their negations, we can show that any finite modification of a finite spectrum is still a finite spectrum. – Chris Eagle Mar 20 '11 at 23:33
@Chris: It does matter if one is to present an explicit solution. – Bill Dubuque Mar 20 '11 at 23:40
@Chris: In any Language. That is you can make it up – Santiago C. Mar 20 '11 at 23:41
up vote 8 down vote accepted

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.

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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".

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I know it exists, but I don't know an example of such a formula. – Santiago C. Mar 21 '11 at 0:00
@Santiago: It would be helpful to revise your question to make that clear. I will leave the answer since it provides a good reference for other readers. – Bill Dubuque Mar 21 '11 at 0:47

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