The Consistency of the Continuum Hypothesis

by Kurt Gödel, Richard Laver (Foreword), Sam Sloan (Cover Design)

The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory by Kurt Gödel Kurt Gödel (1906-1978) was a logician, mathematician and philosopher. He is regarded as one of the most significant logicians of all time, who's work has had immense impact upon scientific and philosophical thinking in the 20th century. Gödel is best known for his three theorems, which set forth and explain the foundations of mathematics. This book provides the proof for the third of his three theorems. Included is a new foreword by Richard Laver, Professor of Mathematics at the University of Colorado at Boulder, who explains in simple, non-technical terms the basic, underlying ideas behind Gödel's theorems and proofs and why they are relevant and important.

Genres: Mathematics, Logic, Philosophy, Science

88 pages, Paperback

First published September 1, 1940

About the author

Kurt Gödel was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.