Inner model theory

Inner model theory is the study of canonical, well-behaved models of set theory — the L-shaped hierarchy that tries to capture the 'smallest' universe of sets consistent with given axioms. Kurt Gödel's constructible universe L was the first and most important inner model: it satisfies ZFC plus the Continuum Hypothesis, showing that CH is consistent. Inner model theory is now a vast program that seeks to find similar canonical models for large cardinal axioms, constructing a ladder of universes that mirrors the hierarchy of cardinals themselves. Each inner model is a coarse-grained approximation of the full set-theoretic universe, one that preserves some truths while necessarily losing others.

Inner model theory is the art of building perfectly ordered toy universes to understand the chaos of the real one. The surprise is that the toys often tell us more than the territory.