ON EXPANDABILITY OF MODELS OF PEANO ARITHMETIC TO MODELS OF THE ALTERNATIVE SET-THEORY

We give a sufficient condition for a countable model M of PA to be expandable to an omega-model of AST with absolute OMEGA-orderings. The condition is in terms of saturation schemes or, equivalently, in terms of the ability of the model to code sequences which have some kind of definition in (M, omega). We also show that a weaker scheme of saturation leads to the existence of wellorderings of the model with nice properties. Finally, we answer affirmatively the question of whether the intersection of all beta-expansions of a beta-expandable model M is the set RA(M, omega)-the ramified analytical hierarchy over (M, omega). The results are based on forcing constructions.