An introduction to hyperarithmetical functions

Abstract

By functional equation we mean an equation of the form(1) A1 … Aκ = B1 … B1.Here the A's and B's are functions of one variable from and to the natural numbers and FG is the function obtained from F and G by composition, i.e. FG(x) = F(G(x)) for all natural numbers x. We wish to investigate finite systems of functional equations. Now if all the A's and B's of (1) are equal to the identity function I (or all equal to the zero function O), then the equation (1) is satisfied trivially. Thus, in order to make the problem of solvability of systems of equations interesting, we must have some function given which will be held fixed throughout. We take the successor function S to be this given function.