Proper and piecewise proper families of reals

TL;DR

This paper introduces proper and piecewise proper families of reals to address an open question in models of PA, exploring their existence across various cardinalities and their structural properties.

Contribution

It defines new classes of families of reals and investigates their existence, advancing understanding of their role in models of PA and Scott sets.

Findings

Proper families are arithmetically closed with proper quotient Boolean algebras.

Piecewise proper families are unions of chains of proper families.

The paper explores the existence of such families across different cardinalities.

Abstract

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.