Definition
Continuum Hypothesis
The continuum hypothesis (
) claims that no set has a cardinality strictly between those of the natural numbers and the real numbers: In ZFC, this is equivalent to
, where is the smallest uncountable cardinality.
Independence from ZFC
Gödel–Cohen independence theorem
Let ZFC denote Zermelo–Fraenkel set theory with the axiom of choice. If ZFC is consistent, it neither proves nor refutes
: This is independence from these axioms, not from every possible mathematical principle.