Finite Arithmetics

Let FM (A), for A = (ω, . . .), be a family of finite, initial segments of A. One can see FM (A) as a family of finite arithmetics related to the arithmetic A. In the talk I will survey what is known about such arithmetics and their logical properties. The main questions will be the following: What is the complexity of theories of such families? What relations can be represented in such families? What are the possible ways to represent some infinite relations? What are the definabilities between such arithmetics? Especially, we focus our attention on finite arithmetics of the form: FM ((ω, +,×)), FM ((ω,×)), FM ((ω,⊥)), where +, ×, ⊥ are addition, multiplication and coprimality relation, respectively.