The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory¶
Gödel, K. (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Princeton University Press.
Cited by¶
3 citations across 3 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Cardinality
- T4 — Continuum Hypothesis and the limits of ZFC. The Continuum Hypothesis — is \(\mathfrak{c} = \aleph_1\)? — is independent of ZFC
This sourceEstablishes relative consistency Con(ZF) → Con(ZFC + GCH) by constructing the constructible universe L, an inner model in which AC and GCH (hence CH) hold.
- T4 — Continuum Hypothesis and the limits of ZFC. The Continuum Hypothesis — is \(\mathfrak{c} = \aleph_1\)? — is independent of ZFC
- Infinity
- Gödel's (1940) consistency proof for the Continuum Hypothesis
This sourceAnnals of Mathematics Studies 3. Princeton: Princeton University Press, 1940. Establishes relative consistency (Con(ZF) → Con(ZFC+GCH)) via the constructible universe L, and articulates the NBG (Neumann–Bernays–Gödel) class-theoretic foundation.
- Gödel's (1940) consistency proof for the Continuum Hypothesis
- Set and Membership
- Not a proper class — proper classes (the "class of all sets," the "class of all groups") are too large to be sets in ZFC; they are a distinct structural object handled in NBG or Morse-Kelley set theory
This sourceAnnals of Mathematics Studies 3. Princeton: Princeton University Press, 1940. Establishes relative consistency (Con(ZF) → Con(ZFC+GCH)) via the constructible universe L, and articulates the NBG (Neumann–Bernays–Gödel) class-theoretic foundation.
- Not a proper class — proper classes (the "class of all sets," the "class of all groups") are too large to be sets in ZFC; they are a distinct structural object handled in NBG or Morse-Kelley set theory
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