The Consistency of the Continuum Hypothesis

GÖDEL, Kurt

Published by Princeton University Press, Princeton, 1940

About this Item

The Constructible Universe - Origin of Inner-Model Theory and the Consistency of the Continuum Hypothesis. First edition, first printing, and a fine copy with provenance, of Gödel's demonstration that the Continuum Hypothesis and the Axiom of Choice are consistent with the standard axioms of set theory. This brief monograph, issued as number 3 in the Annals of Mathematics Studies, provided the first rigorous proof that two of the most debated principles in modern mathematics cannot be disproved from Zermelo-Fraenkel set theory (ZF). It marks the transition from Hilbert's programme of absolute consistency to the modern method of relative consistency and inaugurated the use of definability and model-theoretic construction in set-theoretic foundations. Provenance: Morton G. White (1917-2016), philosopher and historian of ideas, with his ownership inscription on the front blank. White taught at Harvard before joining the Institute for Advanced Study, where he was a colleague of Gödel. A critic of narrow logical empiricism, he sought a unified account of science and philosophy that rejected the division between analytic and synthetic knowledge. His presence at Princeton during Gödel's most productive years places this copy within the immediate circle that witnessed the birth of modern mathematical logic. Gödel's result arose directly from the situation created by his incompleteness theorems of 1931. Those theorems had shown that any consistent formal system capable of expressing elementary arithmetic contains true propositions that are unprovable within the system and that such a system cannot prove its own consistency. The outcome forced a revision of Hilbert's programme, which had aimed to establish the reliability of mathematics by finitary proof methods. Having demonstrated that absolute proof of consistency was impossible, Gödel turned to relative proof: if a given axiom system is consistent, certain extensions of it will be consistent as well. His Consistency of the Continuum Hypothesis is the most powerful realisation of that idea. The Continuum Hypothesis (CH) had dominated discussions of the infinite since Cantor first posed it in the 1870s. Cantor showed that the natural numbers and the real numbers form sets of different cardinalities but was unable to decide whether any intermediate cardinality exists. Hilbert placed the problem first in his celebrated list of 1900, describing it as a test of the adequacy of the mathematical axioms. Gödel's theorem supplied half of the answer: he proved that CH, together with the Axiom of Choice, cannot be refuted from ZF. The complementary half was provided twenty-three years later by Paul Cohen, who proved that CH cannot be derived from ZF either. Together their results established the independence of CH from the standard axioms of mathematics. Gödel's proof depends on the construction of a new mathematical universe, the constructible universe (L). He defines L as a hierarchy built by transfinite recursion through the ordinals: at each stage ? the level L? consists of all sets definable from earlier stages, and L is the union of all such levels. He shows that L is an inner model of ZF set theory-meaning that all the axioms of ZF hold within it-and that in this model every set is definable. From that definability it follows that every set can be well-ordered, so that the Axiom of Choice holds in L. He further proves that in L every infinite cardinal ? satisfies 2^? = ??, which yields the Generalised Continuum Hypothesis (GCH) and therefore CH itself. The decisive conclusion is that, if ZF is consistent, so is ZF + AC + GCH; consequently neither AC nor CH can be disproved from ZF. The method represented a new way of reasoning about mathematics. Instead of searching for proofs inside an axiomatic system, Gödel constructed a model of that system within which particular statements could be seen to hold. The idea that mathematical theories could be examined by building universes satisfying.

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Bibliographic Details

• Title: The Consistency of the Continuum Hypothesis
• Publisher: Princeton University Press, Princeton
• Publication Date: 1940
• Seller catalogs:
  • Mathematics, Analysis, Algebra