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)

GÖDEL, Kurt

Edité par National Academy of Sciences], [Washington, D.C., 1939

A propos de cet article

Gödel's Constructible Universe and Hilbert's First Problem. 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.

N° de réf. du vendeur 6567

Détails bibliographiques

• Titre: 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)
• Éditeur: National Academy of Sciences], [Washington, D.C.
• Date de parution: 1939
• Édition: First edition.
• Catalogues du vendeur:
  • Computers, Numerical Methods
  • Mathematics, Analysis, Algebra