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Work #1563 · Middle

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

Kurt Gödel
1940 · English
Mathematical monograph (lecture-derived) · 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

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Attribute The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis (Middle)
Time · Extent Infinite
Time · Ontological Status Substantival
Time · Grain Continuous
Time · Freedom Deterministic
Time · Traversability Linear
Time · Dimensionality One
Time · Direction Uni-directional
Space · Extent Infinite
Space · Ontological Status Substantival
Space · Curvature Curved
Space · Dimensionality Three
Space · Locality Local
Matter · Extent Infinite
Matter · Ontological Status Substantival
Matter · Conservation Conserved
Matter · Dimensionality Three
Matter · Locality Local
Observer · Time Instance Single
Observer · Space Instance Single
Observer · Knowledge Extent Mediated
Observer · Knowledge Retainment Total
Observer · Physicality Embodied
Observer · Agency Active
Observer · Number Plural
Observer · Metaphysical Agency Impersonal
Observer · Moral Authority Reason
Observer · Theological Method
Energy · Extent Infinite
Energy · Ontological Status Substantival
Energy · Conservation Conserved
Energy · Dispersibility Irreversible
Information · Ontological Status Substantival
Information · Cosmic Conservation Conserved
Information · Personal Conservation Conserved
Information · Granularity Continuous

Dimension-by-Dimension Evidence

What each work's passages reveal about its stance on each of the six dimensions.

Time

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

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.

Space

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

Institute for Advanced Study, Princeton. The 1938-39 lectures (delivered before his emigration to America was permanent) were the basis for the published monograph.

Matter

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

Single mathematical monograph (~70 pages). Form is technical-mathematical: definitions, theorems, proofs in standard set-theoretic notation.

Observer

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

Middle Gödel. The observer is the logician already famous for the 1931 incompleteness theorems, now turning to set theory's foundational questions.

Energy

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

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.

Information

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

Single lecture-derived monograph plus subsequent editions. The constructible-universe construction has become a central tool in modern set theory.

Internal Tensions

Where each work's argument pulls against itself.

The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis

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.