1. 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)?
2. 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?



Recursive ordinals and the minimal standard model of ZF
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. 


