Show that  and  have the same cardinality. [Hint: You may use the fact that every real num...

Show that  and  have the same cardinality. [Hint: You may use the fact that every real number has a decimal expansion, which is unique if expansions that end in an infinite string of 9’s are forbidden.]

A famous conjecture of set theory, called the continuum hypothesis, asserts that there exists no set having greater cardinality than  and lesser cardinality than . The generalized continuum hypothesis asserts that, given the infinite set A, there is no set having greater cardinality than A and lesser cardinality than . Surprisingly enough, both of these assertions have been shown to be independent of the usual axioms for set theory. For a readable expository account, see [Sm].