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Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is the $m+1$-th admissible ordinal relative to $x$.

Anybody knows the proof? Or where to find it?

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There is a model theoretic proof of the generalization to countable sequences which appears in a paper by Simpson and Weitkamp.

High and low Kleene degrees of coanalytic sets Stephen G. Simpson & Galen Weitkamp Journal of Symbolic Logic 48 (2):356-368 (1983)

I believe that this proof is due to Harrington, based on Jensen's and Friedman's model theoretic proof of Sacks's theorem that every countable admissible ordinal is the least admissible relative to some real.

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    $\begingroup$ Hi Ted, nice to see you here! $\endgroup$
    – Andrés E. Caicedo
    Commented Feb 3, 2013 at 20:31
  • $\begingroup$ Yes, Ted, welcome to MathOverflow. $\endgroup$
    – Joel David Hamkins
    Commented Feb 3, 2013 at 21:11
  • $\begingroup$ Ted, thanks. In Simpon-Weitkamp's paper, it is claimed that Harrington has a model theoretical proof. But where to find it? $\endgroup$
    – 喻 良
    Commented Feb 4, 2013 at 4:14
  • $\begingroup$ I asked Prof. Jensen, when he was in NUS, whether he has a model theoretical proof, or by applying Barwise compactness, of his result. He said no. So I guess Harrington's proof must be highly nontrivial. $\endgroup$
    – 喻 良
    Commented Feb 4, 2013 at 7:14
  • $\begingroup$ Yu, the result is Theorem 4.1 of the Simpson and Weitkamp paper and they include a proof. The proof comes before the attribution to Harrington. $\endgroup$
    – Theodore Slaman
    Commented Feb 5, 2013 at 2:48

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