One of the nice features of the first admissible ordinal after $\omega$, i.e. $\omega\_1^{CK}$, is that it is the collection of ordinals whose order type is that of a computable well-ordering on $\omega$. 


Is a similar thing true for all other admissible sets? Specifically suppose $\alpha$ is an admissible ordinal, $\alpha^+$ is the next admissible after $\alpha$ and $\alpha \leq \gamma < \alpha^+$. Is there always an $\alpha$-computable well-ordering on $\alpha$ of order type $\gamma$? 


If so can someone also provide a reference for the result?



Order type of $\alpha$-computable well-orderings
[Surprisingly, no there isn't!](http://www.sciencedirect.com/science/article/pii/0003484379900251) See also [this paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.101.3332&rep=rep1&type=pdf).


Roughly, if $\alpha$ is sufficiently stable, then the next admissible is much greater than the supremum of the $\alpha$-computable well-orderings of $\alpha$.




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Specifically, the relevant fact is:



> 
> If $R$ is an illfounded binary relation on $\omega\_1$ in $L\_{\omega\_1}$, then there is an infinite $R$-descending sequence $\sigma$ with $\sigma\in L\_{\omega\_1}$.
> 
> 
> 


Quick Lowenheim-Skolem arguments show that if $\alpha$ is "sufficiently like" $\omega\_1$ (I think $++$-stable is enough), the same is true: if $R$ is an illfounded binary relation on $\alpha$ in $L\_{\alpha}$, then there is an infinite $R$-descending sequence $\sigma$ with $\sigma\in L\_\alpha$. It's now not hard to show that for such an $\alpha$, the supremum of the order types of $\alpha$-computable well-orderings of $\alpha$ (I'm denoting this by "$\omega\_1^G(\alpha)$" in a paper I'm writing - I don't think there's standard notation) is not admissible (hence is $<\alpha^+$).


