TITLE: Recursive ordinals and the minimal standard model of ZF
QUESTION [6 upvotes]: Does the minimal standard model of ZF contain all recursive ordinals or is it limited (probably by the proof theoretic ordinal of ZF as I suspect but cannot prove)?
Paul J. Cohen's definition of the minimal standard model for ZF.(http://www.ams.org/journals/bull/1963-69-04/S0002-9904-1963-10989-1/S0002-9904-1963-10989-1.pdf) is iterated through all ordinals. What is the least ordinal it must be iterated up to generate the minimal standard model?

REPLY [6 votes]: The minimal transitive (or synonymously, standard) model of ZF is $L_\alpha$, where the ordinal $\alpha$ is chosen to be smallest such that $L_\alpha\models\text{ZF}$. Since ZF proves that $\omega_1^{CK}$ exist, and the recursive ordinals are absolute between $V$ and $L_\alpha$, it follows that $\omega_1^{CK}<\alpha$. So the answer to question 1 is yes. 
The ordinal $\alpha$ is precisely the number of times you have to iterate Gödel's definable power set operation in order to build the minimal transitive model, and I don't know any simpler definition of $\alpha$ than: the height of the minimal transitive model. 
Meanwhile, $\alpha$ is a fairly small countable ordinal, if it exists, and furthermore countable in $L$. An upper bound for complexity is provided by the fact that there is a $\Delta^1_2$ definable relation on $\mathbb{N}$ with order type $\alpha$, simply by noting that we may define the least $L$-code for such a relation, using at most that complexity.