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Journal of the American Mathematical Society

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

 

 

The $ \Pi\sp 1\sb 2$-singleton conjecture


Author: Sy D. Friedman
Journal: J. Amer. Math. Soc. 3 (1990), 771-791
MSC: Primary 03E45
DOI: https://doi.org/10.1090/S0894-0347-1990-1071116-6
MathSciNet review: 1071116
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Abstract: The real $ {0^\char93 } = {\operatorname{Thy}}\left\langle {L,\varepsilon ,{\aleph _1},{\aleph _2}, \ldots } \right\rangle $ is a natural example of a nonconstructible definable real. Moreover $ {0^\char93 }$ has a definition that is absolute: for some formula $ \phi (x),{0^\char93 }$ 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^\char93 }$ 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^\char93 }$. 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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DOI: https://doi.org/10.1090/S0894-0347-1990-1071116-6
Article copyright: © Copyright 1990 American Mathematical Society