Continuum hypothesis

The continuum hypothesis (abbreviated CH) in Template:W states that is there is no set that has a Template:W which is strictly greater than the cardinality of the set of all Template:Ws and strictly less than the cardinality of the set of all Template:Ws. The CH was formulated in 1878 by Georg Cantor. Within the standard axioms of set theory (with the Template:W) and Template:W, 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 Template:W, 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, Template:Cite book. quote from p. 11

• ... there is Brouwer's Template:W, which is utterly destructive in its results. The whole theory of the $\mathrm{\aleph }$'s greater than ${\mathrm{\aleph }}_1$ 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 Template:W and Template:W'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.
  • Template:W, Template:Cite book