$L-$rank (Kunen exercise) I'm stuck with exercise 5 at page 180, where it asks me to compute explicitly the $L-$rank $\rho(\bigcup x)$ of $\bigcup x$ in terms of $\rho(x)$. You can obviously define $\bigcup x$ from the elements of $x$, but that just implies $\rho(\bigcup x)\leq\rho(x)$. What else can I say? Is it different whether $\rho(x)$ is a limit ordinal or not?
At page 167 it says that "there are often subsets of $L(\beta)$ which are in $L$ but not in $L(\beta+1)$". Would anyone give me an example? Is there any way to estimate $\rho(x)$ in terms of the minimum $\alpha$ s.t. $x\subseteq L(\alpha)$? For example, is it true that if $\alpha$ is a successor ordinal then $\rho(x)$ is too?
 A: For the exercise, I don't think there is anything better you can say.  Note that $\rho ( \{ \varnothing \} ) = 1$, but as $\bigcup \{ \varnothing \} = \varnothing$ we have that $\rho ( \bigcup \{ \varnothing \} ) = 0$.  It is thus possible that $\rho ( \bigcup x ) < \rho ( x )$.  On the other hand, as $\omega = \bigcup \omega$ equality is also possible.
As for the second part, I'll just worry about existence.  Note that if $\mathcal{P}^{\mathbf{L}} ( \omega ) \subseteq L(\alpha)$, it follows that $\mathcal{P}^{\mathbf{L}} ( \omega ) \in L ( \alpha + 1 )$ (being the set of all elements of $L(\alpha)$ which are subsets of $\omega$).  It is easy to show that $\mathcal{P}^{\mathbf{L}} ( \omega ) \notin L ( \alpha )$ for all $\alpha < \omega_1^{\mathbf{L}}$ (since the mapping $\alpha \mapsto L(\alpha)$ is absolute (for transitive models of $\mathsf{ZF}-\mathsf{P}$) we have that $\mathbf{L} \models | L ( \alpha ) | = | \alpha |$ for all $\alpha \geq \omega$, and $\mathbf{L}$ correctly computes the $\mathbf{L}$-ranks of its elements).  Therefore for each $\alpha < \omega_1^{\mathbf{L}}$ there must be an $a \in \mathcal{P}^{\mathbf{L}} ( \omega )$ which does not belong to $L ( \alpha )$ (in particular, some such $a$ does not belong to $L ( \omega+1 )$, even though $a \subseteq L ( \omega )$).
This should indicate that there is no easy way to compute $\rho ( x )$ given the least $\alpha$ such that $x \subseteq L ( \alpha )$.
To briefly go back to the exercise, if $a \in \mathcal{P}^{\mathbf{L}} ( \omega )$ does not belong to $L ( \omega )$, it follows that $a$ must be infinite, and so $\bigcup a = \omega$, so $\rho ( \bigcup a ) = \omega$ regardless of whether $\rho ( a )$ was a limit or a successor ordinal.
