An axiomatic basis for computer programming
Communications of the ACM
Arithmetic, first-order logic, and counting quantifiers
ACM Transactions on Computational Logic (TOCL)
FM-Representability and beyond
CiE'05 Proceedings of the First international conference on Computability in Europe: new Computational Paradigms
Fundamenta Informaticae - Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk
Potential Infinity and the Church Thesis
Fundamenta Informaticae - Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk
Theories of initial segments of standard models of arithmetics and their complete extensions
Theoretical Computer Science
Fundamenta Informaticae - Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk
Potential Infinity and the Church Thesis
Fundamenta Informaticae - Topics in Logic, Philosophy and Foundations of Mathematics and Computer Science. In Recognition of Professor Andrzej Grzegorczyk
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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 P2 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 P2 being the set of primes and squares of primes, given by Bès and Richard, 1998.