Geodesic
The Π¹₂-singleton conjecture
Journal of the American Mathematical Society, Tome 03 (1990) no. 4, pp. 771-791

Voir la notice de l'article provenant de la source American Mathematical Society

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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Friedman, Sy D. The Π¹₂-singleton conjecture. Journal of the American Mathematical Society, Tome 03 (1990) no. 4, pp. 771-791. doi: 10.1090/S0894-0347-1990-1071116-6

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