The Banach-Tarski Paradox

This theorem, which assumes the Axiom of Choice, says that certain objects in 3-D space can be partitioned into sets that can be reassembled to form other objects. For example, a sphere can be divided into 5 subsets that can then form two identical spheres. There's a pretty good video explaining this although some details are swept under the rug. See also the Wikipedia page.

The general version of the theorem; any two subsets of 3-D space that are bounded and have nonempty interior (i.e. contain a ball) can be divided into some number of pieces such that Xi is congruent to Yi. There's a proof by Madeline Tremblay that uses only basic linear algebra.

The Continuum Hypothesis

See my description of this.