An inner model M is a transitive class satisfying ZFC, contained in V. Gödel's L (constructible sets) is the canonical inner model; it satisfies GCH, the axiom of choice, and V=L. Other inner models (HOD, L[0#], etc.) capture different set-theoretic properties. Relative consistency is proved by embedding statements into inner models: if M ⊨ φ for a statement φ and M ⊆ V, then Con(ZFC) implies Con(ZFC + φ).
Define L recursively: L₀ = ∅, L_{α+1} = Def(L_α), and L_λ = ⋃_{α < λ} L_α, where Def denotes definable subsets. Prove L ⊨ ZFC. Show Con(ZFC) → Con(ZFC + CH) via the canonical inner model. Explore other inner models and their properties.
From your study of the constructible universe L, you know that Gödel built a specific model inside ZFC — a class where every set is "definable from earlier sets" in a precise transfinite construction. An inner model generalizes this idea: it is any transitive class M satisfying all ZFC axioms and contained within the set-theoretic universe V. Transitivity is the key closure condition: if a ∈ M and b ∈ a, then b ∈ M. This means the model "doesn't import aliens" — every element of every element of M is already in M. Transitivity ensures that the ∈-relation inside M agrees with the actual ∈-relation in V, making M a "genuine" universe of sets, not an artificial simulation.
The key technique inner models provide is relative consistency: showing that if ZFC is consistent, then ZFC + φ is also consistent for some additional statement φ. The method is to exhibit an inner model M where φ holds. Gödel showed L ⊨ V=L (every set is constructible), L ⊨ AC (the axiom of choice holds), and L ⊨ GCH (the generalized continuum hypothesis). Since L is a definable class inside any model of ZFC, if ZFC is consistent then L gives a model of ZFC + GCH. This proves Con(ZFC) → Con(ZFC + GCH): you cannot derive a contradiction from ZFC + GCH without first deriving one from ZFC. In other words, GCH cannot be *disproved* from ZFC alone.
Notice what the technique does *not* prove: it does not show GCH is *true*, or that V = L. It shows only that assuming GCH leads to no new contradictions. This is the nature of relative consistency — a conditional, not an absolute, result. The complementary result, that GCH cannot be *proved* from ZFC, came later from Cohen's forcing method, which constructs models where GCH fails. Together, the inner model technique and forcing establish the independence of GCH from ZFC: neither the hypothesis nor its negation follows from the axioms. The inner model Gödel constructed answers "can GCH be false?"; forcing answers "can GCH be true by necessity?" — and both answers are no.
Beyond L, the inner model landscape is rich. HOD (hereditarily ordinal-definable sets) is the class of all sets whose entire membership tree is definable from ordinals — strictly larger than L, but still an inner model satisfying AC. The core model program (due to Jensen and others) generalizes L to accommodate large cardinal axioms: if large cardinals exist in V, L is too small to capture them, but a larger inner model built from "mice" (small structured sets with embeddings) can. The existence of the inner model L[0#] — L extended by a "sharp" encoding structural information about L — is equivalent to 0# existing, which in turn follows from certain large cardinal assumptions. Each inner model is a lens for studying which set-theoretic truths are already "built in" to ZFC versus which require additional axioms. Inner models are not alternative realities to be chosen between — they are tools for probing the structure of V itself.