  We compare two naturally arising notions of unknotting number for 2-spheres
in the 4-sphere: namely, the minimal number of 1-handle stabilizations needed
to obtain an unknotted surface, and the minimal number of Whitney moves
required in a regular homotopy to the unknotted 2-sphere. We refer to these
invariants as the stabilization number and the Casson-Whitney number of the
sphere, respectively. Using both algebraic and geometric techniques, we show
that the stabilization number is bounded above by one more than the
Casson-Whitney number. We also provide explicit families of spheres for which
these invariants are equal, as well as families for which they are distinct.
Furthermore, we give additional bounds for both invariants, concrete examples
of their non-additivity, and applications to classical unknotting number of
1-knots.
