Proper and piecewise proper families of reals

I introduced the notions of proper and piecewise proper families of reals to make progress on an open question in the field of models of PA about whether every Scott set is the standard system of a model of PA. A family of reals X is proper if it is arithmetically closed and the quotient Boolean algebra X/fin is a proper poset. A family is piecewise proper if it is the union of a chain of proper families of size $\leq\omega_1$. Here, I explore the question of the existence of proper and piecewise proper families of reals of different cardinalities.