Abstract

I introduced the notions of proper and piecewise proper families of reals to make progress on a long standing open question in the field of models of Peano Arithmetic [5]. A family of reals is proper if it is arithmetically closed and its quotient Boolean algebra modulo the ideal of finite sets is a proper poset. A family of reals is piecewise proper if it is the union of a chain of proper families each of whom has size ≤ ω1.Here, I investigate the question of the existence of proper and piecewise proper families of reals of different cardinalities. I show that it is consistent relative to ZFC to have continuum many proper families of cardinality ω1 and continuum many piecewise proper families of cardinality ω2