### Scott’s problem for Proper Scott sets

Victoria Gitman
Source: J. Symbolic Logic Volume 73, Issue 3 (2008), 845-860.

#### 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.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396751
Digital Object Identifier: doi:10.2178/jsl/1230396751
Mathematical Reviews number (MathSciNet): MR2444272
Zentralblatt MATH identifier: 1161.03024

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