Crashcourse Models in Set Theory I am currently working through the lecture notes. In the end we had a short introduction to Models in Set Theory, but since it was quite to the end, we did not really go into details.

So my questions:
What are Models in Set Theory for? 
When and where do I need them?
And, what 'are' Models in Set Theory?

Best, Luca
 A: One of the most important area of modern set theory is the study of extensions of ZFC, independence and consistency results, and the large cardinals. Models in various sense are crucial to study these results.
First to address what are models in set theory. These are structures (in the model theory sense) over a first order language which includes some binary relation symbol $E$. The language may include other symbols (which are useful for instance in the formulation of indescribability). Sometimes these structures may be sets for instance $H(\kappa)$, $V_\kappa$, or $L_\kappa$. Sometimes proper classes with a definable $E$ relation may also be called structures or models. Among these are the $V$, $WF = \bigcup_{\alpha \in Ord} V_\alpha$, $L$, and various inner models. There are technical difficulties with these structures. For example, the satisfaction relation is not definable uniformly in all formulas. However, they are often used in consistency results by relativization. (See Kunen for the logical aspect of proper class models.)
Models are important in consistency results. For instance the inner model $L$ establishes to consistency of $GCH$, $AC$, $V= L$, $\Diamond$, etc. In forcing, countable transitive models of ZFC (or finite fragments of ZFC) are extended to other countable transitive models $M[G]$ satisfying various other statements. For instance, the consistency of $\neg CH$ is proved by forcing. (Again refer to Kunen about the technical aspect of whether countable transitive models of ZFC exists and what the generic extension $M[G]$ is.)
Models are also very important in the study of large cardinal axioms. For instance, if $\kappa$ is an inaccessible cardinal, then $V_\kappa$ is a model of $ZFC$. (By the incompleteness Theorem, $ZFC$ can not prove the large cardinal axioms.) Many of the other large cardinals are defined using various notions of elementary embedding between models and critical points. For instance, a measureable cardinal is the critical point of nontrivial elementary embedding of $V$ into a transitive inner model. 
Inner Model Theory is an important area of set theory that produced various consistency results using inner models. They are often used to relate consistency strength of various statements to large cardinals. Notable examples are the consistency of various forms of determinacy from Woodin cardinals. In particular, the axiom of determinacy is equiconsistent with infinitely many Woodin cardinals. 
