Scott's problem for proper Scott sets
Victoria Gitman
TL;DR
Under the assumption of PFA, this paper proves that every proper Scott set corresponds to the standard system of a model of PA, linking set-theoretic properties with models of arithmetic.
Contribution
It establishes a new connection between proper Scott sets and models of PA assuming PFA, extending understanding of their structure and properties.
Findings
Proper Scott sets are standard systems of models of PA under PFA.
Properness of Scott sets relates to their arithmetical closure and Boolean algebra quotient.
The result depends on the Proper Forcing Axiom (PFA).
Abstract
I show that assuming PFA, every proper Scott set is the standard system of a model of PA. A Scott set X is proper if it is arithmetically closed and the quotient Boolean algebra X/Fin is a proper partial order.
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Taxonomy
TopicsAdvanced Algebra and Logic · semigroups and automata theory · Advanced Topology and Set Theory