The assertion that the union of any Aleph_1 many measure zero sets is still measure zero. This is independent of ZFC. Of course, it implies the failure of the Continuum Hypothesis, but is not equivalent to this.
There are a huge variety of such statements in the field known as Cardinal Characteristics of the Continuum. For example, what is the additivity of the meager ideal (the ideal of all meager sets)? It is at least Aleph_1, but can be larger. What is the smallest size of a family of functions f:omega to omega such that every function is bounded by an element of the family? It can be Aleph_1, or larger independently. There are dozens of such examples.
