Church's thesis as an empirical hypothesis

The studies of Church’s thesis (further denoted as CT) are slowly gaining their intensity. Since different formulations of the thesis are being proposed, it is necessary to state how it should be understood. The following is the formulation given by Church: [CT] The concept of the effectively calculable function is identical with the concept of the recursive function. The conviction that the above formulation is correct, finds its confirmation in Church’s own words: We now define the notion, [...] of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers [...]. Frege thought that the material equivalence of concepts is the approximation of their identity. For any concepts F, G: Concept F is identical with concept G iff ∀x (Fx ≡ Gx) . The symbol Fx should be read as “object x falls under the concept F”. The contemporary understanding of the right side of the above equation was dominated by the first order logic and set theory.