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I know Jensen developed a forcing notions in $L$ that adds a unique, minimal and $\Delta_3^1$ $L$-generic real. In his paper Definable sets of minimal degree he says that Solovay had already shown the consistency relative to $\mathsf{ZF}$ of $``V$ is the constructible closure of a real which is the unique solution of a $\Pi_2^1$ predicate (hence the real is $\Delta_3^1$)$"$. I searched a bit, but I couldn't find an exhaustive presentation of Solovay's result. My questions are:

  • Is there a paper/thesis where the abovementioned result due to Solovay is explained?
  • Are there other known forcing notions adding an $L$-generic real which is the unique solution of a $\Pi_2^1$ predicate?

EDIT: Solovay's result is fully discussed here, even though it would be nice to find a more modern account. The second question still stands.

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    $\begingroup$ I think he might just be referring to zero sharp and the model $L[0^\#]$. The fact that zero sharp is a $\Pi^1_2$-singleton is proved in both Jech and Kanamori’s books. $\endgroup$
    – Gabe Goldberg
    Commented Apr 13, 2023 at 21:03
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    $\begingroup$ @GabeGoldberg That's what I initially thought, but he isn't. I edited the question and added a link to the paper, written by Jensen and Solovay, in which they apply the almost disjoint forcing to get such a real (from $\mathsf{ZFC}$ alone). $\endgroup$
    – Lorenzo
    Commented Apr 14, 2023 at 14:05
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    $\begingroup$ Interesting! I didn't know about this paper. $\endgroup$
    – Gabe Goldberg
    Commented Apr 14, 2023 at 14:41
  • $\begingroup$ Is this question related to your previous deleted question about the failure of perfect set property for $\Pi^1_2$-sets but PSP holds for $\Sigma^1_2$-sets? $\endgroup$
    – Hanul Jeon
    Commented Apr 17, 2023 at 5:01

1 Answer 1

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Since the first question has already been answered in the EDIT at the end of the question, I will focus on the second question.

The short answer to second question is: Yes, a good source is Sy Friedman's paper The $\Pi^1_2$-conjecture, Journal of the American Mathematical Society, Vol. 3, No. 4 (Oct., 1990), pp. 771-791.

In the above paper, Friedman negatively settled a conjecture of Robert Solovay, a conjecture that stated: Assuming the existence of $0^\sharp$ , there are no $\Pi^1_2$ singletons $s$ such that $s \in \mathrm{L}[0^\sharp]$ and yet $0^\sharp \notin \mathrm{L} [s]$. Friedman's construction is labyrinthine; a key ingredient of it is Jensen's technique of Coding the Universe, and the important related work of René David in his paper A very absolute $\Pi^1_2$ real singleton, Ann. Math. Logic 23 (1982), no. 2-3, 101–120 (1983).

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