The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, *every* (nontrivial) $X$ is "almost" the jump of something. There are a number of extensions and variations of the Posner-Robinson theorem; I'm interested in two, due to unpublished work of Hugh Woodin.


In multiple places - e.g. page 208 of "Proceedings of the 13th Asian Logic Conference" - we find the claim that Woodin proved the following higher-order analogues of the Posner-Robinson theorem:


* If $X$ is not hyperarithmetic, then there is some $G$ such that $X\oplus G\equiv\_T \mathcal{O}^G$.
* (Assume $X^\#$ exists for every real $X$.) If $X$ is not constructible, then there is some $G$ such that $X\oplus G\equiv\_T G^\#$.


My question is: are these proofs available anywhere?



Woodin on Posner-Robinson for the hyperjump and sharp

> 
> [MR2449474 (2009j:03067)](http://www.ams.org/mathscinet-getitem?mr=2449474) Woodin, W. Hugh. *A tt version of the
>  Posner-Robinson theorem*. Computational prospects of infinity. Part II.
>  Presented talks, 355–392, Lect. Notes Ser. Inst. Math. Sci. Natl.
>  Univ. Singap., 15, World Sci. Publ., Hackensack, NJ, 2008.
> 
> 
> 


The proof is nice, invoking both recursion-theoretic and set-theoretic tools. Hugh uses a Prikry-like forcing notion, and considers forcing over countable non-standard $\omega$-models of (a large fragment of) set theory. 


(For other examples of forcing over non-standard models, see *Projective prewellorderings vs projective wellfounded relations* by X. Shi, or the last few chapters of the monograph **Super-real fields** by Hugh and Garth Dales.) 


