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Set Theory and the Continuum Hypothesis

Paul J. Cohen
Dover Publications
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

If ever a book deserved to be described as a "classic," this is it. It contains Paul Cohen's account of his proof that the Continuum Hypothesis (CH) is independent of standard Zermelo-Frankel set theory. His methods show the same for the Axiom of Choice (AC). Both CH and AC had been shown to be consistent with standard set theory by Gödel in the 1940s. Cohen's contribution was to show that their negations are also consistent, so that both statements are independent of the standard axioms.

After much argument in the early 20th century, most mathematicians have been content to include the Axiom of Choice with the standard collection of axioms for set theory. This has not been the case, however, for the Continuum Hypothesis. In fact, in the conclusion to this book Cohen argues (on informal grounds, of course) that mathematicians should take CH as false.

The Dover edition includes an article by Cohen entitled "My Interaction with Kurt Gödel" and an introduction by Martin Davis. Unfortunately, the main text has not been reset; it is still a photographic reproduction of typed notes.

No matter. No self-respecting library could be without this one.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.


General Background in Logic
Zermelo-Fraenkel Set Theory
The Consistsency of the Continuum Hypothesis and the Axiom of Choice
The Independence of the Continuum Hypothetis and the Axiom of Choice