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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 !1. 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 addition- ally arithmetically closed. I also investigate the question of the existence of proper Scott sets. In this paper, I use the Proper Forcing Axiom (PFA) to make partial progress on a half-century-old question in the folklore of models of Peano Arithmetic about whether every Scott set is the standard system of a model of PA. The Proper Forcing Axiom is a generalization of Martin's Axiom that has found application in many areas of set theory in recent years. I will show that, assuming PFA, every arithmetically closed Scott set whose quotient Boolean algebra X/Fin is proper is the standard system of a model of PA. I will begin with some technical and historical details. We can associate to every model M of PA a certain collection of subsets of the natural numbers called its standard system, in short SSy(M). The natural numbers N form the initial segment of every model of PA. The standard system consists of sets that arise as intersections of the definable (with parameters) sets of the model with the standard part N. One thinks of the standard system as the traces left by the definable sets of the model on the natural numbers. A dierent way of characterizing sets in the standard system uses the notion of coding. Let us say that a set A N is coded in a model M if M has an element a such that (a)n = 1 if and only if n 2 A. Here, (a)x can refer to any of the reader's favorite methods of coding with elements of a model of PA, e.g., by defining (a)x as the x th digit in the binary expansion of a. It is easy to see that for nonstandard models we can equivalently define the standard system to be the collection of all subsets of the natural numbers coded in the model. What features characterize standard systems? Without reference to models of arithmetic, a standard system is just a particular collection of subsets of the natural numbers. Can we come up with a list of elementary (set theoretic, computability theoretic, etc.) properties that X P (N) must satisfy in order to be the standard system of some model of PA? The notion of a Scott set encapsulates three key features of standard systems. Definition 1.1. X P (N) is a Scott set if |
