Geodesic
On the coprimeness relation from the viewpoint of monadic second-order logic
Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1225-1239.

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Let $\mathfrak{C}$ denote the structure of the natural numbers with the coprimeness relation. We prove that for each non-zero natural number $n$, if a $\Pi^1_n$-set of natural numbers is closed under automorphisms of $\mathfrak{C}$, then it is definable in $\mathfrak{C}$ by a monadic $\Pi^1_n$-formula of the signature of $\mathfrak{C}$ having exactly $n$ set quantifiers. On the other hand, we observe that even a much weaker version of this property fails for certain expansions of $\mathfrak{C}$.
Keywords: coprimeness, monadic second-order logic, definability, weak arithmetics. 
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S. O. Speranski; F. N. Pakhomov. On the coprimeness relation from the viewpoint of monadic second-order logic. Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1225-1239. http://geodesic.mathdoc.fr/item/IM2_2022_86_6_a9/

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