  The projective hierarchy can be extended into the transfinite as follows: For a set X, let Def(X) consist of the subsets of X that are definable over X using parameters from X. This is the definable powerset operation. The hierarchy L(ℝ) is the result of starting with ℝ and iterating the definable powerset operation along the ordinals; more precisely, we let L0(ℝ) = Vω+1, Lα+1(ℝ) = Def(Lα(ℝ)), Lλ(ℝ) = ⋃α&lt;λLα(ℝ) for limit ordinals λ, and, finally, 