# Continuum hypothesis

hypothesis that no set has a cardinality between that of the integers and that of the real numbers

The continuum hypothesis (abbreviated CH) in set theory states that is there is no set that has a cardinality which is strictly greater than the cardinality of the set of all integers and strictly less than the cardinality of the set of all real numberss. The CH was formulated in 1878 by Georg Cantor. Within the standard axioms of set theory (with the axiom of choice) and first order logic, under the assumption that the standard axioms are logically consistent, Kurt Gödel proved in 1938 that the CH cannot be disproved and Paul Cohen proved in 1963 that the CH cannot be proved.

## Quotes

• In the Realist philosophy, one wholeheartedly accepts traditional mathematics at face value. All questions such as the Continuum Hypothesis are either true or false in the real world despite their independence from the various axiom systems. The Realist position is probably the one that most mathematicians would prefer to take.
• Paul Cohen, "Comments on the foundations of set theory". Axiomatic Set Theory, Part 1. American Mathematical Soc.. 31 December 1971. pp. 9–16. ISBN 9780821802458; ed. by Dans S. Scott . quote from p. 11
• ... there is Brouwer's intuitionism, which is utterly destructive in its results. The whole theory of the ${\displaystyle \,\aleph \,}$ 's greater than ${\displaystyle \,\aleph _{1}\,}$  is rejected as meaningless (Brouwer 1907, 569). Cantor's conjecture itself receives several different meanings, all of which, though very interesting in themselves, are quite different from the original problem. They lead partly to affirmative, partly to negative answers (Brouwer, 1907, I: 9; III: 2). Not everything in this field, however, has been sufficiently clarified. The “semi-intuitionistic” standpoint along the lines of H. Poincaré and H. Weyl ... would hardly preserve substantially more of set theory.
• Kurt Gödel, What is Cantor's Continuum Problem?, November 1947, vol. 54, pp. 515–525, American Mathematical Monthly, reprinted on pages 470– 485 in Benacerraf and Putnam's collection Philosophy of Mathematics, 2nd ed., Cambridge University Press, 1983. quote from p. 473 & p. 474
• Those who argue that the concept of set is not sufficiently clear to fix the truth-value of CH have a position which is at present difficult to assail. As long as no new axiom is found which decides CH, their case will continue to grow stronger, and our assertion that the meaning of CH is clear will sound more and more empty.
• Donald A. Martin, "Hilbert's first problem: the continuum hypothesis". Mathematical developments arising from Hilbert problems, Part 1. Symposium in Pure Mathematics, Northern Illinois University, 1974. Proceedings of symposia in pure mathematics; volume 28. American Mathematical Society. 1976. pp. 81–92; edited by Felix E. Browder