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?



Countable admissible ordinals
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.


