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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



The $\Pi ^ 1_ 2$-singleton conjecture

Author: Sy D. Friedman
Journal: J. Amer. Math. Soc. 3 (1990), 771-791
MSC: Primary 03E45
MathSciNet review: 1071116
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Abstract: The real ${0^\# } = {\operatorname {Thy}}\left \langle {L,\varepsilon ,{\aleph _1},{\aleph _2}, \ldots } \right \rangle$ is a natural example of a nonconstructible definable real. Moreover ${0^\# }$ has a definition that is absolute: for some formula $\phi (x),{0^\# }$ is the unique real $R$ such that $L[R] \vDash \phi (R)$. Solovay conjectured that there is a real $R$ such that $0{ < _L}R{ < _L}{0^\# }$ and $R$ also has such an absolute definition. We prove his conjecture by constructing a $\Pi _2^1$-singleton $R$, $0{ < _L}R{ < _L}{0^\# }$. A variant of our construction produces a countable nonempty $\Pi _2^1$ set of reals not containing a $\Pi _2^1$-singleton. The latter result answers a question of Kechris.

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Article copyright: © Copyright 1990 American Mathematical Society