An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which is the smallest possible inner model, built by transfinitely iterating the eight basic Gödel operations (or, alternatively, the “definable” or “restricted” powerset). $L$ is the most studied inner model, as it has many combinatorial and fine-structural properties. It admits natural generalisations, $L(A)$ and $L[A]$.
However, by a theorem of Scott, measurable cardinals cannot exist within $L$. So inner model theory is the study of trying to create inner models, which can accommodate large cardinals and have similar structure to $L$. I believe I am close to understanding many of these constructions. However, there are some evidently crucial aspects that I do not quite understand the purpose or necessity of.
From what I’ve read, in all of modern inner model theory, these inner models are built out of “premice”, structures of the form $J_\alpha^A \models \mathrm{ZFC}^-$ (where the $J$-hierarchy is an alternate way of building up the model $L[A]$), which have to have well-founded ultrapowers (in which case they are called mice), and one needs a comparison lemma for establishing which mice are initial segments of (ultrapowers of) each other. This is all well and good, but I have three questions:
- What’s the purpose of building the desired inner model, which is a proper class satisfying all of $\mathrm{ZFC}$, out of set-sized fragments not necessarily satisfying Powerset? Why not directly construe it as $L[A]$? Perhaps I’m misunderstanding.
- What’s the motivation or intuition behind using iterated ultrapowers, and especially, why must our mice have well-founded ones?
- Why do we need to be able to compare the mice which we use to build our model, i.e. why do we need an algorithm for determining whether one mouse is an initial segment of an iterated ultrapower of another?
My expectation is that this is all down to making it easier to prove that the resulting model in fact has fine structure, but I don’t see how. I read some of Steel’s “The Comparison Lemma” in my quest for an answer, and it mentioned that for $L$ the mice are just well-founded structures satisfying a fragment of $\mathrm{ZFC}$ and $V = L$, which was also helpful but the intuition is still nowhere near fully built for me.
