Standard systems and the Scott set problem

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The Scott set problem

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?

The origins of the problem goes back the paper of Scott [1], where it is shown that the answer is positive for countable sets $\mathfrak X$.

Knight and Nadel [2], and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.

Jim Schmerl [3] gave positive answer for arithmetic closures of sets of (arithmetic) Cohen generics of any cardinality.

For more recent attempts employing forcing axioms see [4] and [5].


Kanovei's question

Is there a Borel model $M\models PA$ whose standard system is the power set of $\omega$?


Woodin's question

If $\mathfrak X$ is a Borel Scott set, is there a Borel model $M\models PA$ whose standard system is $\mathfrak X$?

References

  1. Dana Scott. Algebras of sets binumerable in complete extensions of arithmetic. Proc. Sympos. Pure Math., Vol. V, pp. 117--121, Providence, R.I., 1962. MR   bibtex
  2. Julia Knight and Mark Nadel. Models of arithmetic and closed ideals. J. Symbolic Logic 47(4):833--840 (1983), 1982. www   DOI   MR   bibtex
  3. James H. Schmerl. Peano models with many generic classes. Pacific J. Math. 46:523--536, 1973. MR   bibtex
  4. Fredrik Engström. A note on standard systems and ultrafilters. J. Symbolic Logic 73(3):824--830, 2008. www   DOI   MR   bibtex
  5. Victoria Gitman. Scott's problem for proper Scott sets. J. Symbolic Logic 73(3):845--860, 2008. www   DOI   MR   bibtex
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