Is there an example of a sentence with spectrum {p|p is prime}?
I may need to resort to some theorem of Algebra, but not sure if it will help.
Thanks
2026-04-13 19:44:38.1776109478
L-sentences and spectra
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Yes, I believe there is, though I won't claim to be able to show it.
I'm not very knowledgeable about the subject, but from I gather, there is a full characterization of first-order spectra in terms of complexity of characteristic functions. It says that spectra are exactly the sets decidable by a non-deterministic Turing machine in $O(2^{cx})$ time, where $c$ is (any) constant; in particular, any set decidable in polynomial time is a spectrum. Due to AKS, this includes the set of prime numbers.
To read more on the subject, see this survey on arXiv or this monograph on Mostowski, or the paper containing the original result (behind a paywall).