The $\Pi ^ 1_ 2$-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 ${0^\# } = {\operatorname {Thy}}\left \langle {L,\varepsilon ,{\aleph _1},{\aleph _2}, \ldots } \right \rangle$ 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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*An absolute*$\Pi _2^1$

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Article copyright:
© Copyright 1990
American Mathematical Society