Scott’s problem for Proper Scott sets

Abstract

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of PA. Two decades later, Knight and Nadel extended his result to Scott sets of size ω₁. Here, I show that assuming the Proper Forcing Axiom (PFA), every A-proper Scott set is the standard system of a model of PA. I define that a Scott set 𝔛 is proper if the quotient Boolean algebra 𝔛/Fin is a proper partial order and A-proper if 𝔛 is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.