Editing Georg Cantor (section)

====Continuum hypothesis====
{{Main|Continuum hypothesis}}
Cantor was the first to formulate what later came to be known as the [[continuum hypothesis]] or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is ''exactly'' aleph-one, rather than just ''at least'' aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to [[mathematical proof|prove]] it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.<ref name="daub280" />

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by [[Kurt Gödel]] and a 1963 one by [[Paul Cohen (mathematician)|Paul Cohen]] together imply that the continuum hypothesis can be neither proved nor disproved using standard [[Zermelo–Fraenkel set theory]] plus the [[axiom of choice]] (the combination referred to as "[[ZFC]]").<ref>Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these is [[W. Hugh Woodin]]. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.</ref>