Intuition behind extenders

An extender is a system of compatible ultrafilters. The compatibility manifests itself directly: We have projections from ultrafilters on "larger" spaces to those on "smaller" spaces, or natural ways of extending "small" ones into "larger" ones. But, more relevantly, the compatibility means that the elementary embeddings you can form by taking the corresponding ultrapowers "fit together". Thus, by taking directed limits of the corresponding embeddings (which is why the first kind of compatibility is needed), we can describe embeddings that we could not capture as ultrapowers by a single ultrafilter.

A useful intuitive way of seeing where these systems of ultrafilters are coming from is to imagine an embedding $j:V\to M$ is given. You can derive an ultrafilter on its critical point as usual. You can then form the ultrapower by that ultrafilter, say $i:V\to N$, and there is a natural way to re-embed $N$ back into $M$, say $k:N\to M$ so the relevant diagram commutes: $k\circ i = j$. However, $k$ needs not be the identity, so it has a critical point, and we can consider the derived ultrafilter, and iterate this procedure for as long as needed. (For example, it may well be that if $\kappa$ is the critical point of $j$, then $j(\kappa)$ is much larger than $i(\kappa)$.)

This intuitive description falls a bit short of capturing the full power of extenders, as I am only looking at "linear systems" of ultrapowers this way. But extenders allow us to capture embeddings where the target model is too "thick" to be obtained by this linear process. A $(\kappa,\lambda)$-extender derived from an embedding $j$ then provides us with an approximation to $j$. For example, the $(\kappa,\kappa+1)$-extender in the example from the previous paragraph would just give us $i$ and $N$. But the larger $\lambda$ is, the closer this approximation is to $j$, though we may need a proper class extender to capture $j$ completely sometimes.

However, the definition is very flexible, so it can be applied in situations where $M$ needs not have strong closure properties.

[Now, extenders (and pre-extenders) can be defined by describing directly the compatibility properties of a system of ultrafilters, so we do not need to start with the ultrapowers, just as we can describe (normal) ultrafilters directly, without needing to appeal to a given embedding. This is useful, as it allows us to use extenders from a model $V'$ to form ultrapowers in another model $V''$, as long as $V'$ and $V''$ have some sufficiently large initial segment in common.]