Does there exist a strictly increasing and continuous function of ordinal numbers whose
smallest critical number (i.e. fixed point) is the smallest non-constructive ordinal
number (in the sense of Church and Kleene)? If so, does there exist such a function all
of whose critical numbers (i.e. fixed points) are non-constructive and admissible ordinal numbers?



Some questions about functions of Ordinal Numbers
Any admissible ordinal is the limit of indecomposable ordinals. Pick an $\omega$-sequence of these ordinals, $\alpha\_0<\alpha\_1<\dots$ with limit $\omega\_1^{CK}$, so the order type of $[\alpha\_i,\alpha\_{i+1}]$ is $\alpha\_{i+1}$. Now define $f$ on $[0,\alpha\_0]$ by $f(\beta)=\alpha\_0+1+\beta$, on $(\alpha\_0,\alpha\_1]$ by $f(\beta)=\alpha\_1+\beta$, on $(\alpha\_1,\alpha\_2]$ by $f(\beta)=\alpha\_2+\beta$, etc. Then define for $\beta\ge\omega\_1^{CK}$, $f(\beta)=\beta$. This $f$ is normal, and has $\omega\_1^{CK}$ as its first fixed point. Note that the fact that $\omega\_1^{CK}$ is admissible is irrelevant here; in fact, all that matters here is that it has countable cofinality, and is indecomposable. 


If instead we want $f(\alpha)$ to be admissible for each $\alpha$, pick a club through the admissible ordinals, and let $f$ be its increasing enumeration. Suppose for instance that $\kappa$ is strongly inaccessible, and note that then there is a club of $\alpha<\kappa$ such that $V\_\alpha\prec V\_\kappa$, but each such $\alpha$ is of course admissible. We can easily relax the assumptions, to turn this into an argument that there is indeed, provably in $\mathsf{ZF}$, a (proper class) club of admissible ordinals. Note, on the other hand, that a limit of admissible ordinals need not be admissible, so some care is needed.


