Some extensions of the principles of idealization transfer and choice in the relative internal set theory

Abstract

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let α be a level and let Φ(x, y) be anαexternalαbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by α, corresponds an elementy x such that Φ(x, y x ) holds, then there exists a function of choice ψ such that, which is a very general principle of choice, for everyx dominated by α, Φ(x, ψ(x)) holds. More than that, we establish that if all the elementsy x are uniformly dominated by a level β then we can prescribe that the function of choice is also dominated by β.