GÖDEL, Kurt.

The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis. [With:] Consistency-Proof for the Generalized Continuum-Hypothesis. Offprints from the Proceedings of the National Academy of Sciences 24 (December 1938) and 25 (April 1939).

[Washington, D.C: National Academy of Sciences], 1938; 1939.

Together these two offprints contain Gödel’s first announcement and sketch of one of the central achievements in modern logic: the construction of his inner model of the “constructible universe” and the resulting proof that, if the usual axioms of Zermelo–Fraenkel set theory (ZF) are consistent, then so are the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH). This was the first substantial progress on Hilbert’s first problem since its formulation in 1900, and the result that framed all subsequent discussion of the continuum problem until Paul Cohen’s invention of forcing a quarter of a century later. More broadly, Gödel’s work showed for the first time that fundamental set-theoretic statements could be neither provable nor disprovable from the accepted axioms—an insight that fundamentally reshaped the nature of mathematical foundations. His construction demonstrated that within ZF one can define a canonical inner universe of “constructible” sets which satisfies all the axioms of ZF and, in addition, AC and GCH. In other words, AC and GCH cannot introduce a contradiction if none already exists in ZF: they are relatively consistent. These offprints are thus the printed starting point of the modern inner-model tradition in set theory and the essential complement to Cohen’s later independence work.

Provenance: The 1939 offprint is an author’s presentation copy, bearing Gödel’s handwritten inscription “With the compliments of the author / Kurt Gödel” on the front wrapper. Within the text there is one small pencil annotation by a contemporary, technically informed reader: a neat marginal note reading “gödel says” beside Theorem 6, accompanied by a pencilled equivalence sign clarifying the logical step Gödel asserts at that point. This single annotation reflects close, expert engagement with the argument rather than casual reading. No other marks of ownership or annotation are present. The set thus stands as a rare, directly author-associated witness to Gödel’s foundational work during his years at the Institute for Advanced Study.

In the first, shorter note of December 1938, Gödel gives an almost disarmingly compact statement of his main theorem. Let be a standard system of axioms for set theory—von Neumann’s system or Fraenkel’s axioms, with the Axiom of Choice omitted. Gödel shows that if is consistent, then so is the theory obtained by adding as new axioms (1) the Axiom of Choice and (2) the Generalized Continuum Hypothesis, together with two additional propositions about the existence of certain linear non-measurable sets and their complements. The proof is only sketched, but Gödel already signals the crucial idea: he constructs, “on the basis of the axioms of ,” a canonical model consisting of all “mathematically constructible” sets, organized in a ramified hierarchy. If a contradiction could be derived from the extended axioms, he explains, it would manifest as a contradiction inside this constructible model itself—hence the new axioms are relatively consistent with so long as is.

The 1939 offprint Consistency-Proof for the Generalized Continuum-Hypothesis expands this sketch into a more substantial, though still compressed, exposition and introduces the notation and basic ideas that would later be fully elaborated in his 1940 monograph The Consistency of the Continuum Hypothesis—Gödel’s only book-length publication in logic, and now recognised as one of the great foundational texts of twentieth-century mathematics. In these offprints Gödel first defines the transfinite sequence of levels of constructible sets: beginning with , taking at each successor stage all sets definable over the previous level with parameters, and taking unions at limit stages. The union is the universe of constructible sets. Working entirely inside ZF, Gödel shows that is a transitive inner model—closed under membership and containing all ordinals—in which every axiom of ZF holds when relativized to . This in itself was an innovation: no one before Gödel had constructed a fully fledged inner model of ZF. He then demonstrates that the definability constraints governing ensure that every set of reals has the regularity properties needed for the combinatorial argument yielding GCH to go through, and that a definable well-ordering of the continuum exists, thereby establishing AC and GCH in .

The philosophical and mathematical significance of this cannot be overstated. The continuum hypothesis, first formulated by Cantor in the 1870s, asserts that there is no cardinal strictly between the integers and the real numbers. Cantor believed the continuum possessed a determinate, discoverable size—indeed, he regarded the arithmetic of infinite cardinals as one of the central foundations of mathematics. When Hilbert placed CH as the first of his twenty-three problems in 1900, he intended it to symbolise the new era of axiomatisation: one should be able to resolve questions about the continuum by purely logical means. The introduction of Zermelo’s axioms in 1908, later expanded by Fraenkel and Skolem, seemed to promise such a foundation. But Gödel’s incompleteness theorems of 1931 had already shaken this aspiration. They showed that any sufficiently rich axiomatic system could not prove its own consistency, and that there would always be true statements unprovable within it.

Gödel’s results of 1938–39 added a new dimension: they demonstrated that one could not hope to disprove CH or AC from ZF alone. These axioms, contested by philosophers and mathematicians alike during the interwar period, were shown to be fully compatible with the standard axioms of set theory—so long as those axioms were consistent. In the long view, Gödel’s work represents the moment when the relationship between axioms and mathematical truth shifted irrevocably: no longer could foundational questions be expected to have unique answers within a single canonical framework. Instead, one had to consider mathematical universes—models—in which different principles hold. Gödel provided the first such canonical model.

The constructible universe , introduced here for the first time in print, soon became—and remains—the baseline inner model in set theory. As Kanamori and Moore have emphasized, Gödel’s construction reshaped the conceptual foundations of set theory: it rendered precise the cumulative-hierarchy intuition, according to which sets are built in successive stages, and it enshrined definability as the decisive criterion for set existence within . Over subsequent decades, became the reference universe against which the strength of new axioms would be calibrated: large cardinals, determinacy principles, and many finer combinatorial hypotheses all derive their philosophical and mathematical meaning in part from how they behave relative to . Indeed, the entire “core model” programme—an immense body of work analysing canonical inner models with large cardinals—is unimaginable without Gödel’s initial construction.

Gödel emigrated from Vienna to the Institute for Advanced Study in Princeton in 1933, aided in part by Albert Einstein, who took a personal interest in ensuring that the young logician—already recognised as a once-in-a-generation intellect—could escape the dangers of rising National Socialism. Einstein and Gödel became close friends; they walked home together daily from the Institute, and Einstein later remarked that he went to his office “only to have the privilege of walking home with Gödel.” This unusual friendship was rooted in a shared fascination with the philosophical implications of mathematics and physics, and in Einstein’s admiration for Gödel’s unparalleled logical precision. It was within this remarkably supportive environment—one shaped decisively by Einstein’s assistance—that Gödel developed the ideas culminating in the constructible universe and the offprints offered here.

Yet Gödel’s results were incomplete in a tantalising way: while they showed that CH could hold in a canonical inner universe, they did not show that CH must hold in every model of set theory. The possibility remained that its negation might also be consistent. That possibility was realized dramatically by Paul Cohen in 1963–64 with the invention of forcing. Cohen constructed models of ZFC in which CH fails and models of ZF in which AC fails. Cohen himself described Gödel’s work as the indispensable first half of the independence story: inner models show that certain axioms cannot be disproved; forcing shows they cannot be proved. Together, Gödel and Cohen established the modern understanding that CH is independent of ZFC—an answer to Hilbert’s problem that neither Hilbert nor Cantor could have anticipated.

These two offprints record the precise moment at which Gödel first introduced the constructible universe and set the stage for the modern independence revolution. They represent the conceptual bridge between Cantor’s vision of the infinite, Hilbert’s programme of axiomatic certainty, and the pluralistic model-theoretic landscape that characterises contemporary set theory. They are also the essential precursors to Gödel’s 1940 monograph The Consistency of the Continuum Hypothesis (the work represented by item no. 6476 on the Sophia Rare Books site). That monograph is the complete and definitive presentation of the ideas first sketched here: it substantially expands the proofs, clarifies the construction of the hierarchy , refines the relativisation of axioms to the constructible universe, and offers Gödel’s mature philosophical reflections on axiomatisation and the nature of mathematical truth. Where the offprints are compact and programmatic, the monograph is systematic and authoritative, and for that reason the two offprints constitute the indispensable historical and conceptual prelude to the 1940 book.

In retrospect, Gödel’s two preliminary announcements stand not simply as early formulations but as the point at which the continuum problem entered its modern phase. They contain, in compressed form, the conceptual machinery that reshaped the foundations of set theory and prepared the ground for every subsequent development in the study of definability, inner models, and independence. Taken together with the later 1940 monograph that completed their programme, they chart the moment when Cantor’s hierarchy of infinities and Hilbert’s axiomatic vision converged into a deeper understanding of mathematical possibility—an understanding defined not by a single universe of sets but by the diversity of models in which central questions may have different answers. No other documents capture this turning point with greater immediacy or authority.

References: Gödel, The Consistency of the Continuum Hypothesis, 1940. Gödel, Collected Works, vols. I–II, 1986–1990. Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel, 1997. Kanamori, The Higher Infinite, 2003. Kanamori, Gödel and Set Theory, Bulletin of Symbolic Logic 14 (2008), pp. 351–378. Moore, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence, 2013. Hilbert, Mathematical Problems, 1900. Maddy, Believing the Axioms, Journal of Symbolic Logic 53 (1988), pp. 481–511 & 559–577. Cohen, The Discovery of Forcing, Rocky Mountain Journal of Mathematics 32 (2002), pp. 1071–1100.

Two offprints, large 8vo (258 x 174 mm). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis (offprint from Proceedings of the National Academy of Sciences 24, December 1938, pp. 556–557): 2 pp., published without wrappers as issued. Consistency-Proof for the Generalized Continuum-Hypothesis (offprint from Proceedings of the National Academy of Sciences 25, April 1939, pp. 220–224): 5 pp., in the original tan printed wrapper; front wrapper inscribed in Gödel’s hand “With the compliments of the author / Kurt Gödel.” A single contemporary pencil annotation (“gödel says” with an equivalence sign) appears in the margin beside Theorem 6. Both offprints exceptionally well preserved, with only minor handling to extremities; the 1939 wrapper clean and unfaded, inner margins sound throughout.

Item #6567

Price: $75,000.00