Coprimality in Finite Models

Abstract

We investigate properties of the coprimality relation within the family of finite models being initial segments of the standard model for coprimality, denoted by $\mathrm{FM}((\omega,\bot))$.

Within $\mathrm{FM}((\omega,\bot))$ we construct an interpretation of addition and multiplication on indices of prime numbers. Consequently, the first order theory of $\mathrm{FM}((\omega,\bot))$ is Π$^{\rm 0}_{\rm 1}$–complete (in contrast to the decidability of the theory of multiplication in the standard model). This result strengthens an analogous theorem of Marcin Mostowski and Anna Wasilewska, 2004, for the divisibility relation.

As a byproduct we obtain definitions of addition and multiplication on indices of primes in the model $(\omega,\bot,\leq_{P_2})$, where P ₂ is the set of primes and products of two different primes and ≤ _X is the ordering relation restricted to the set X. This can be compared to the decidability of the first order theory of $(\omega,\bot,\leq_P)$, for P being the set of primes (Maurin, 1997) and to the interpretation of addition and multiplication in $(\omega,\bot,\leq_{P^2})$, for P ² being the set of primes and squares of primes, given by Bès and Richard, 1998.