  We are interested in the sets of (k-tuples of) reals appearing in this hierarchy, that is, P((ωω)k) ∩ Lα(ℝ) for α ∈ On. Notice that in the definition of the projective sets, complementation corresponds to negation and projection corresponds to existential quantification over real numbers. It follows that the projective sets of (k-tuples of) reals are exactly P((ωω)k) ∩ L1(ℝ). Thus, the hierarchy of sets P((ωω)k) ∩ Lα(ℝ) (for α ∈ On) does indeed yield a transfinite extension of the projective sets. 