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Is the question of the downward density of the w-REA sets under $\leq_a$ still open? If not can anyone point me to a proof? That is do we know if for every $\omega$-REA set $X >_a 0_a$ there exists another $\omega$-REA $Y$ with $X >_a Y >_a 0$?

I presume it is still open and it's an interesting question IMO but before wasting any time on it I figured I'd ask a bit more widely to be sure.

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Probably still open. James Barnes' dissertation (2018) addresses initial segments under the arithmetic reducibility, but is not specifically about $\omega$-CEA degrees.

Barnes, James S., On the decidability of the $\Sigma_2$ theories of the arithmetic and hyperarithmetic degrees as uppersemilattices, J. Symb. Log. 82, No. 4, 1496-1518 (2017). ZBL1391.03031.

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    $\begingroup$ I'm pretty surprised that anyone knows what is in my dissertation. $\endgroup$
    – James
    Commented Feb 21, 2019 at 13:04
  • $\begingroup$ @James did you think about the $\omega$-CEA question at all? $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Feb 21, 2019 at 15:28
  • $\begingroup$ I did not. The questions of initial segments (or, dually, density) were answered by you and Shore for the hyperdegrees and M. Simpson (not Steve) for the arithmetic degrees. I worked on the "other end" determining when we can extend embedding. $\endgroup$
    – James
    Commented Feb 21, 2019 at 21:56

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