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Knight and Nadel [2], and independently Ehrenfeucht (unpublished), gave positive answer for $\mathfrak X$ of cardinality $\aleph_1$.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