GENERAL PRINCIPLE OF CHOICE IN THE RELATIVE INTERNAL SET-THEORY

The framework of this Note is RIST (see [2]), an extension of the classic set theory ZFC. We have established that the principle of choice of ZFC can be extended to a large class of formulas of RIST. Let alpha be a level. If Phi is a formula of the language of RIST, we prove that if Phi is any (alpha)external and (alpha)bounded formula, if corresponding to each element x dominated by alpha there is an element y(x), such that Phi (x, y(x)) holds, then there exists a function of choice psi such that for any x dominated by alpha, Phi (x, psi(x)) holds, which is a very general principle of choice. We have proved also: a) that for any level mu dominated by alpha, this principle still works for a subclass of the class of the (mu)external (alpha)bounded formulas, b) that, under the condition that the y(x) are uniformly dominated by a level beta which dominates alpha, we can prescribe that the choice function psi is dominated by beta also.