When do we need ZFC models to be transitive? I learn set theory mainly through Kunen's classical book: set theory, an introduction to independence proofs. In that book, Almost all models are transitive. I checked some basic lemmas and notions, if the models are not transitive, they may be fail or not absolute.
However, in other books or lectures, authors mainly talk about "models" rather than "transitive models".
Does this mean models need not to be transitive there? Or just omitted "transitive"? Or identify models with their transitive collapsing？I can't guess which one is right.
Thanks for any help.
 A: This is too long for a comment, so:
One exception that should be mentioned is that of small elementary substructures of large, transitive sets.
Fix a regular uncountable cardinal $\kappa$ and let $H_\kappa=\{x:|\text{trcl}(x)|<\kappa\}$. As you probably know if you've been reading Kunen, $H_\kappa$ satisfies ZFC-P, where P is the Power Set Axiom (also, $H_\kappa\models$ ZFC iff $\kappa$ is inaccessible).
Now, take an elementary substructure $M\prec H_\kappa$ with $M=\aleph_0$. These objects are extremely useful in set theory. A certain tension exists between $M$ being small, yet "knowing a lot" of information from the larger structure. This can be exploited to prove many different things. For example, the proof that GCH holds in L uses countable elementary substructures, and a lot facts in infinitary combinatorics (like Fodor's Lemma, the $\Delta$-system Lemma, etc) can be given alternative proofs using countable elementary substructures. In fact, many of the proofs using countable elementary substructures are much slicker, and feel almost magical. See also the proof of Arhangelskii's Theorem for an application to point-set topology.
After that sales pitch, a reasonable question to ask is whether such an $M$ can be transitive? The answer is a resounding no, whenever $\kappa>\aleph_1$. Indeed, for such $\kappa$, $\omega_1\in H_\kappa$. Moreover, one can show that
$$
H_\kappa\models \omega_1\text{ is the least uncountable cardinal}
$$
So that $\omega_1$ is definable without parameters in $H_\kappa$. By elementarity, $\omega_1\in M$. Now, if $M$ were transitive, then $\omega_1\subseteq M$, so $M$ is uncountable, contradiction.
Collapsing these $M$ is also part of the game, and that yields some important information (in the $L$ case, the collapse must be of the form $L_\delta$ for some countable $\delta$), but that "tension" I mentioned before is lost. For example, $\omega_1\in M$ but $\omega_1$ does not belong to the collapse. Thus, whatever ordinal the collapse thinks is the first uncountable cardinal, is not the real $\omega_1$.
