Forcing names, parameters in definitions, and the Iterative Conception of Set So, I've been trying to learn as much as I can about forcing. I know that a model provides its own (trivial) forcing extension. What I'm curious about is whether there is a way to think of the iterative hierarchy in terms of (trivial?) forcing extensions?
The iterative hierarchy for $\mathsf{ZF}$ can be given in the traditional manner:

The first level, $V_0$, is defined to be the empty set, $\emptyset$
  (if the set theory is impure, this is also where the
  urelements, i.e., the non-set individuals, reside). Subsequent
  levels are formed by taking the powerset of the previous stage such
  that the level immediately following $V_n$ (for finite ordinals $n$),
  the successor level $V_{n+1}$, is defined as
  $\mathcal{P}(V_n)$. Once you have formed all of the finite stages, you
  form the first limit level $V_\omega$ by taking the infinite union of the preceding levels, $\bigcup_{n<\omega} V_n$. The process then repeats for the
  successor levels of $V_\omega$, $V_{\omega + 1}, V_{\omega +
 1},\dots$, with the successor levels of $V_\omega$ unionized to form
  the next limit level $V_{\omega \cdot 2}$. The universe of
  sets, $V$, is the union of all the levels: $V = \bigcup_{\alpha \in
 O} V_\alpha$, where $O$ is the class of all ordinals.

Is there a way to understand the various levels as forcing extensions of preceding levels? I'm asking because it's common, in talk of the iterative conception, to allow that sets formed at previous levels can be used as parameters in defining new sets. These parameters sort of reminded me of the names that get introduced when you add constants for your forcing language.
Is there any interesting connection between the levels of $\mathsf{ZF}$ and the forcing extensions of a theory? Can we understand the levels as something akin to forcing extensions of preceding levels?
 A: Not really - they are fundamentally different types of extension. One of the crucial properties of forcing is that a forcing extension $V\subset V[G]$ is an end extension - no set "gets new elements" - but not a top extension (forcing adds no new ordinals). A top extension is one in which every new set has higher rank than every old set - that is, exactly as $V_{\alpha+1}$ is to $V_\alpha$. So in a sense, forcing is really "orthogonal" to the cumulative hierarchy - iterating powersets builds "up," forcing builds "sideways.

Building off a comment below: note that nontrivial forcing extensions are never elementary extensions. Now, iterating powerset doesn't yield elementary extensions level-by-level either - that is, $V_\alpha$ is never an elementary substructure of $V_{\alpha+1}$ (since $V_{\alpha+1}$ satisfies "there is a set $X$ of maximal rank," so must $V_\alpha$, so $\alpha=\beta+1$; but then $V_\alpha$ and $V_\beta$ disagree on what the largest ordinal is!) - but we can have $V_\alpha\prec V_\beta$ for certain $\alpha<\beta$.
To find such $\alpha$ and $\beta$: Fix $\alpha_0$. Now by iteratively closing under Skolem functions, we can find an $\alpha>\alpha_0$ such that $V_\alpha\prec V$. Fix $\beta_0>\alpha$, and similarly find $\beta>\beta_0$ such that $V_\beta\prec V$. Then we use the general fact from model theory that $$A\preccurlyeq C, B\preccurlyeq C, A\subseteq B\implies A\preccurlyeq B.$$
(This argument actually takes more than ZFC - in particular, as written it implies the consistency of ZFC! - but not too much more.)
