The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory

Gödel's central claim is that the axiom of choice and the generalized continuum hypothesis are consistent with the axioms of Zermelo-Fraenkel set theory, assuming those axioms are consistent. The monograph rejects the assumption that these principles must be either self-evident or refutable; it shows instead that they are independent of the standard axioms in the sense that they cannot be disproved from them. Gödel proves this by constructing an inner model—the constructible universe L—in which both principles hold, and by arguing that if a contradiction could be derived from them, it could be derived in L and hence from the standard axioms alone. The result is a relative consistency proof, not a proof of the principles themselves.

The constructible universe L is a hierarchy of sets built in stages indexed by the ordinals. At each stage, instead of taking all subsets of the previous stage, Gödel admits only those subsets that are definable (with parameters) over the previous stage. This restriction yields a 'thin' universe that satisfies the standard axioms of set theory but in which every set is definable and well-ordered. Because L is definable within the standard universe, one can prove inside set theory that L satisfies the axiom of choice and the generalized continuum hypothesis. If the standard axioms are consistent, then L is a model of them, and since L also satisfies AC and GCH, those principles are consistent with the standard axioms .

The monograph was published in 1940 by Princeton University Press as Number 3 in the Annals of Mathematics Studies series . Gödel had emigrated to the United States in 1940, and the work was written during a period of transition in his career. The date is significant because it marks the publication of a result that Gödel had announced in 1938, in a brief note in the Proceedings of the National Academy of Sciences . The 1940 monograph is the full exposition of the consistency proof, and it became the standard reference for the constructible universe and for the method of inner models in set theory. Its publication by Princeton University Press reflects Gödel's growing association with the Institute for Advanced Study.

Gödel's proof shows that AC and GCH are consistent with the standard axioms; Paul Cohen's 1963 results show that they are independent, i.e., that their negations are also consistent. Cohen developed the method of forcing to construct models of set theory in which the axiom of choice fails, or in which the continuum hypothesis fails, thereby complementing Gödel's inner-model construction with an outer-model technique . Together, the two results establish the independence of AC and GCH from Zermelo-Fraenkel set theory: neither the principles nor their negations can be proved from the standard axioms. Cohen's work, for which he received the Fields Medal, builds directly on the framework that Gödel's monograph established, and it is impossible to understand the independence results without understanding the constructible universe.

The axiom of constructibility (V = L) asserts that every set is constructible—that the universe of sets is exactly the constructible universe L. Gödel does not adopt this axiom as a new axiom of set theory; he uses it as an internal hypothesis to prove that if L is the whole universe, then AC and GCH hold. The axiom of constructibility is stronger than the axiom of choice: it implies AC, but AC does not imply V = L. Gödel was cautious about V = L, recognizing that it is a restrictive hypothesis that settles many questions in set theory (such as the continuum hypothesis) but may not be true of the intended universe of sets. The monograph's consistency proof shows that V = L is consistent with the standard axioms, not that it is true.

The monograph was immediately recognized as a major contribution to set theory and logic. Gödel's consistency proof resolved the most important open question in set theory—whether the axiom of choice and continuum hypothesis could be refuted from the standard axioms—and it introduced the method of inner models, which remains a fundamental tool in set theory and in proof theory. The constructible universe L became a central object of study, and the monograph's techniques were extended and refined by later set theorists, including Jensen, who developed the fine structure of L. The work also influenced the philosophy of mathematics, because it showed that set theory is not a completed science but a field in which fundamental questions can be settled by relative consistency proofs .

A common misreading treats the consistency proof as establishing the truth of the axiom of choice or the continuum hypothesis. Gödel's result is a relative consistency proof: it shows that if the standard axioms of set theory are consistent, then adding AC and GCH does not introduce a contradiction. It does not show that AC and GCH are true, nor that they are the only consistent options. Cohen's later work showed that the negations of AC and GCH are also consistent, so the question of which principles are true remains open. The proof's significance lies in its method—the constructible universe and inner models—rather than in any claim about the truth of the principles it concerns.