The constructible universe and relative consistency

For a set A, let Def(A) contain the subsets of A defined by first-order formulas over the structure with domain A and membership restricted to A, using finitely many parameters from A. Satisfaction for this set structure is definable by recursion on formulas. This does not contradict Tarski's theorem: it is not a truth predicate for the entire set-theoretic universe. The formula code and finite parameter tuple can be collected as a set, and Separation and Replacement produce the family of defined subsets.

Set L₀ to the empty set, L at a successor to Def of the preceding stage, and L at a nonzero limit to the union of the earlier stages. Transfinite recursion gives each set stage, and their union over all ordinals defines the class L. The stages are transitive and increasing, and L contains every ordinal. This description does not turn L into a set. In particular, it does not offer a counterexample to the earlier proof that no set contains all ordinals.

Gödel's theorem establishes that L satisfies ZFC and GCH in the relativized sense. More precisely, for each axiom of ZFC, ZF proves its translation restricting quantifiers to L; ZF also proves the relativized GCH statement. Verifying this includes the difficult closure and definability work needed for Replacement, Power Set, a canonical well-order, and cardinal counting. It is not proved merely by observing that L is transitive. Here the theorem is cited as a proved primary-source result, while the finite-level calculations illustrate its definition.

An inner model's power set of A contains its own subsets of A, not automatically every subset available outside. For A in L, that internal power set is the intersection of the ambient power set with L. The inner-model theorem ensures that this collection is itself represented in L. It need not equal Def(A), which uses only parameters and quantification over A itself. Likewise L can compute an ordinal's cardinality differently from the ambient universe even though ordinal membership and ordinal order are shared.

If a contradiction had a proof from ZFC plus GCH, replacing each set-theoretic quantifier by its L restriction would translate that finite proof into a contradiction from ZF. This explains the consistency implication without requiring an externally well-founded model. If one instead begins externally with an actual model M of ZF, its internally interpreted constructible universe gives the corresponding model-theoretic argument. M may be ill-founded from outside; demanding a transitive model would add an assumption not supplied by mere consistency.

Assuming ZFC is consistent, its second incompleteness theorem prevents it from proving its own consistency by presenting a set model of all of ZFC. There is no conflict with proving each axiom relativized to the proper class L: the class is not a set structure whose complete satisfaction relation has been asserted internally. Nor does the inner-model theorem prove V=L. That equation is an additional principle, compatible with ZFC if ZFC is consistent, but not an instruction to identify every ambient subset with a constructible one.