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| − | A Scott set is a set of sets of natural numbers $\mathfrak X$ such that $(\omega, {\mathfrak X})\models {\sf WKL}_0$, Assume $\lnot{\sf CH}$. Is every Scott set the standard system of a nonstandard model of PA?
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| − | The origins of the problem goes back the paper of Scott <cite> scott1962:algebras </cite>, where it is shown that the answer is positive for countable sets $\mathfrak X$.
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| − | Knight and Nadel <cite> knightnadel1982:models </cite>, and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.
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| − | Jim Schmerl <cite> schmerl1973:peano </cite> gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality.
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| − | For more recent attempts employing forcing axioms see <cite> gitman2008:scott </cite>
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| − | {{References}}
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