Scott’s problem for proper Scott sets

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V. Gitman, “Scott’s problem for proper Scott sets,” J. Symbolic Logic, vol. 73, no. 3, pp. 845–860, 2008.

Some 40 years ago, Dana Scott proved that every countable Scott set is the standard system of a model of ${\rm PA}$. Two decades later, Knight and Nadel extended his result to Scott sets of size $\omega_1$. Here, I show that assuming the Proper Forcing Axiom (${\rm PFA}$), every A-proper Scott set is the standard system of a model of ${\rm PA}$. I define that a Scott set $\mathfrak{X}$ is proper if the quotient Boolean algebra $\mathfrak{X}/\text{fin}$ is a proper partial order and A proper if $\mathfrak{X}$ is additionally arithmetically closed. I also investigate the question of the existence of proper Scott sets.