The assertion that if
"If a set X is smaller in cardinality than another set Y, then the power set P(X) is smaller X has fewer subsets than P(Y)Y."
Althought the statement sounds obvious, it is actually independent of ZFC. This is true under that The statement follows from the Generalized Continuum Hypothesis, which is consistent with but there are models of ZFC having counterexamples, but it even in relatively concrete cases, where X is also consistent with ZFC that the continuum P(omega) natural numbers and Y is equinumerous with P(Aleph_1)a certain uncountable set of real numbers (but nevertheless the powersets P(X) and P(Y) can be put in bijective correspondence). This situation occurs under Martin's Axiom, among numerous other possibilitieswhen CH fails.
