THE CONSISTENCY STRENGTH OF PROJECTIVE ABSOLUTENESS

It is proved that in the absence of proper class inner models with Woo din cardinals, for each n epsilon {1,...,omega}, Sigma(3+n)(1) absolut eness (i.e., the stability of the Sigma(3+n)(1) theory of the reals un der set forcing in a strong sense) implies there are n strong cardinal s in K (where this denotes a suitably defined global version of the co re model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency stre ngth of Sigma(3+n)(1) absoluteness is exactly that of n strong cardina ls so that in particular projective absoluteness is equiconsistent wit h the existence of infinitely many strong cardinals. It is also argued how this theorem is to be construed as the first step in the long ran ge program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle po sitively a conjecture of Woodin and thereby solve the last Delfino pro blem.