The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis
Gödel's 1940 monograph on the relative consistency of AC and GCH with ZF set theory
Tradition: Mathematical logic / foundations of mathematics / set theory
Gödel's 1940 monograph — relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with ZF set theory
Published by Princeton University Press in 1940 (Annals of Mathematics Studies 3) from Gödel's 1938-39 lectures at the Institute for Advanced Study, 'The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory' proves that the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) cannot be disproved from the Zermelo-Fraenkel axioms (ZF), if the ZF axioms are themselves consistent. The proof's technical core is the construction of the 'constructible universe' L (or 'L' for Latin 'constructibilis') — the inner model obtained by restricting set-theoretic construction to sets definable from sets of lower rank by first-order formulas with parameters. Gödel proves: (1) L is an inner model of ZF (the construction can be carried out within ZF, and L satisfies all the ZF axioms); (2) L satisfies the Axiom of Choice; (3) L satisfies the Generalized Continuum Hypothesis. Therefore, if ZF is consistent, then ZF + AC + GCH is also consistent (since L, an inner model of ZF, satisfies ZF + AC + GCH). The result was foundational for twentieth-century set theory: it established relative consistency and (combined with Paul Cohen's 1963 independence proof for the negations of AC and GCH) showed that both AC and CH are independent of ZF. The constructible-universe construction has become a central tool in modern set theory; the L hierarchy has been continuously generalised (Jensen's fine-structural analysis, the modern inner-model program).
Author
Editions cited
- The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Princeton University Press, Annals of Mathematics Studies 3, 1940)
- Revised editions: 1951, 1968, 1990 (with new preface)
- Reprinted in Kurt Gödel, Collected Works, Vol. II: Publications 1938-1974, ed. Solomon Feferman et al. (Oxford, 1990)
- Critical commentary: Akihiro Kanamori, The Higher Infinite (Springer, 2nd ed. 2003); Saharon Shelah, Cardinal Arithmetic (Oxford, 1994)
School Embodiments
Major contribution to mathematical logic and set theory.
"L is an inner model of ZF + AC + GCH." (Consistency of AC and GCH, ch. III-IV)
Foundational paper for the analytic-philosophical treatment of set-theoretic ontology.
"The constructible hierarchy as inner model." (Consistency of AC and GCH)
Gödelian-Platonist background — though formally a consistency proof, animated by realist intuitions about sets.
"The constructible sets are themselves real mathematical objects." (Gödel's realist gloss)
Structural-set-theoretic methodology.
"Inner-model construction yields the desired structure." (Consistency of AC and GCH)
Mathematical realism about sets.
"Sets are objective mathematical objects." (Consistency of AC and GCH — Gödelian view)
Mathematical-naturalist methodology.
"Standard set-theoretic methods." (Consistency of AC and GCH)
Internal Tensions
Gödel's major set-theoretic contribution; the constructible-universe construction remains a central tool. Together with Cohen's 1963 independence proof for the negations of AC and GCH, it definitively established the independence of these axioms from ZF — answering a question Cantor had posed in the 1870s and Hilbert had placed first on his 1900 list of mathematical problems.
I. Time
1940. Gödel was 33; he had emigrated from Vienna to Princeton in early 1940 after escaping Nazi-occupied Europe via the Trans-Siberian Railway and Pacific crossing.
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II. Space
Institute for Advanced Study, Princeton. The 1938-39 lectures (delivered before his emigration to America was permanent) were the basis for the published monograph.
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III. Matter
Single mathematical monograph (~70 pages). Form is technical-mathematical: definitions, theorems, proofs in standard set-theoretic notation.
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IV. Observer
Middle Gödel. The observer is the logician already famous for the 1931 incompleteness theorems, now turning to set theory's foundational questions.
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V. Energy
Foundational-mathematical energies. The book is one of the major twentieth-century contributions to set theory, comparable in importance to Zermelo's 1908 axiomatisation and Cohen's 1963 independence proof.
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VI. Information
Single lecture-derived monograph plus subsequent editions. The constructible-universe construction has become a central tool in modern set theory.
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How The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis resolves each dilemma
34 resolved positions across 4 dimensions, including 6 distinctive where the majority of schools go the other way · 23 unaligned.
Each dimension is sorted so minority positions come first. Mainstream positions are folded into an expandable list.
Time · 9 dilemmas · 5 distinctive
Persistence, the future, and the direction of becoming.