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Knight and Nadel [2], and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.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

- 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@incollection {scott1962:algebras,

AUTHOR = {Scott, Dana},

TITLE = {Algebras of sets binumerable in complete extensions of arithmetic},

BOOKTITLE = {Proc. Sympos. Pure Math., Vol. V},

PAGES = {117--121},

PUBLISHER = {American Mathematical Society},

ADDRESS = {Providence, R.I.},

YEAR = {1962},

MRCLASS = {02.72},

MRNUMBER = {0141595 (25 \#4993)},

MRREVIEWER = {H. Ribeiro},

}

- Julia Knight and Mark Nadel.
*Models of arithmetic and closed ideals.* J. Symbolic Logic 47(4):833--840 (1983), 1982. www DOI MR bibtex@article {knightnadel1982:models,

AUTHOR = {Knight, Julia and Nadel, Mark},

TITLE = {Models of arithmetic and closed ideals},

JOURNAL = {J. Symbolic Logic},

FJOURNAL = {The Journal of Symbolic Logic},

VOLUME = {47},

YEAR = {1982},

NUMBER = {4},

PAGES = {833--840 (1983)},

ISSN = {0022-4812},

CODEN = {JSYLA6},

MRCLASS = {03C62 (03C50 03D30)},

MRNUMBER = {683158 (85d:03072)},

MRREVIEWER = {S. S. Goncharov},

DOI = {10.2307/2273102},

URL = {http://dx.doi.org/10.2307/2273102},

}

- James H. Schmerl.
*Peano models with many generic classes.* Pacific J. Math. 46:523--536, 1973. MR bibtex@article {schmerl1973:peano,

AUTHOR = {Schmerl, James H.},

TITLE = {Peano models with many generic classes},

JOURNAL = {Pacific J. Math.},

FJOURNAL = {Pacific Journal of Mathematics},

VOLUME = {46},

YEAR = {1973},

PAGES = {523--536},

ISSN = {0030-8730},

MRCLASS = {02H20},

MRNUMBER = {0354351 (50 \#6831)},

MRREVIEWER = {M. Boffa},

}

- Fredrik Engström.
*A note on standard systems and ultrafilters.* J. Symbolic Logic 73(3):824--830, 2008. www DOI MR bibtex@article {engstrom2008:anote,

AUTHOR = {Engstr{\"o}m, Fredrik},

TITLE = {A note on standard systems and ultrafilters},

JOURNAL = {J. Symbolic Logic},

FJOURNAL = {Journal of Symbolic Logic},

VOLUME = {73},

YEAR = {2008},

NUMBER = {3},

PAGES = {824--830},

ISSN = {0022-4812},

CODEN = {JSYLA6},

MRCLASS = {03C62 (03C20 03C57)},

MRNUMBER = {2444270 (2009i:03030)},

DOI = {10.2178/jsl/1230396749},

URL = {http://dx.doi.org/10.2178/jsl/1230396749},

}

- Victoria Gitman.
*Scott's problem for proper Scott sets.* J. Symbolic Logic 73(3):845--860, 2008. www DOI MR bibtex@article {gitman2008:scott,

AUTHOR = {Gitman, Victoria},

TITLE = {Scott\'s problem for proper Scott sets},

JOURNAL = {J. Symbolic Logic},

FJOURNAL = {Journal of Symbolic Logic},

VOLUME = {73},

YEAR = {2008},

NUMBER = {3},

PAGES = {845--860},

ISSN = {0022-4812},

CODEN = {JSYLA6},

MRCLASS = {03C62 (03E65)},

MRNUMBER = {2444272 (2009f:03047)},

MRREVIEWER = {Roman Kossak},

DOI = {10.2178/jsl/1230396751},

URL = {http://dx.doi.org/10.2178/jsl/1230396751},

}

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