Inner Models and Relative Consistency Proofs

Core Idea

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).

How It's Best Learned

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.

Common Misconceptions

• Assuming inner models are 'true' (they are one model among many; V may not equal L).
• Confusing relative consistency with resolution (GCH is consistent with and independent of ZFC, so neither is 'correct').

Explainer

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.