Theorem

Cohen's Independence Theorem

Logic and Foundations

Cohen's Independence Theorem shows, using the technique of forcing that he invented for the purpose, that the continuum hypothesis cannot be proved from the standard Zermelo-Fraenkel axioms of set theory together with the axiom of choice. Named for Paul Cohen, who proved it in 1963, it combines with Kurt Godel's earlier proof that the continuum hypothesis cannot be disproved from those same axioms to establish that the continuum hypothesis is genuinely independent of them, neither provable nor refutable.

Facts

Statement

Using forcing, neither the continuum hypothesis nor the axiom of choice can be proved from the Zermelo-Fraenkel axioms of set theory. 1

Classification

Statement Form

Impossibility Theorem 1

Connections

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Impossibility Theorem

In Branch

Set Theory

Source Paul Cohen (Wikipedia)

Sources

1. Paul Cohen (Wikipedia)

Career
Cohen is noted for developing a mathematical technique called forcing, which he used to prove that neither the continuum hypothesis (CH) nor the axiom of choice can be proved from the standard
In Branch: Set Theory, Lead paragraph
Paul Joseph Cohen (April 2, 1934, March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo-Fraenkel set theory, for which he was awarded a Fields Medal.
In Group: Set Theory, Lead section
the axiom of choice are independent from Zermelo-Fraenkel set theory, for which he was awarded a Fields Medal.

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Cohen's Independence Theorem

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