The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small"
infinite set of integers and the "large" infinite set of real
numbers
(the "continuum"). Symbolically, the continuum
hypothesis is that .
Problem 1a of Hilbert's problems asks if the
continuum hypothesis is true.
Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel
set theory. However, using a technique called forcing,
Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation
of the continuum hypothesis was added to set theory.
Together, Gödel's and Cohen's results established that the validity of the continuum
hypothesis depends on the version of set theory being
used, and is therefore undecidable (assuming the
Zermelo-Fraenkel axioms together with
the axiom of choice).
Conway and Guy (1996, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also undecidable:
is
for every ?
The continuum hypothesis follows from generalized continuum hypothesis, so .
Woodin (2001ab, 2002) formulated a new plausible "axiom" whose adoption (in addition to the Zermelo-Fraenkel axioms
and axiom of choice) would imply that the continuum
hypothesis is false. Since set theoreticians have felt for some time that the Continuum
Hypothesis should be false, if Woodin's axiom proves to be particularly elegant,
useful, or intuitive, it may catch on. It is interesting to compare this to a situation
with Euclid's parallel postulate more than
300 years ago, when Wallis proposed an additional axiom that would imply the parallel
postulate (Greenberg 1994, pp. 152-153).
and the "large" infinite set of real
numbers 
(the "continuum"). Symbolically, the continuum
hypothesis is that 
.
Problem 1a of Hilbert's problems asks if the
continuum hypothesis is true.
for every 
?
The continuum hypothesis follows from generalized continuum hypothesis, so 
.
