The consistency strength of projective absoluteness

Abstract

It is proved that in the absence of proper class inner models with Woodin cardinals, for each n ϵ {1,…, ω}, ∑ 3 + n 1 absoluteness (i.e., the stability of the ∑ 3 + n 1 theory of the reals under set forcing in a strong sense) implies there are n strong cardinals in K (where this denotes a suitably defined global version of the core model for one Woodin cardinal as exposed by Steel. Combined with a forcing argument of Woodin, this establishes that the consistency strength of ∑ 3 + n 1 absoluteness is exactly that of n strong cardinals so that in particular projective absoluteness is equiconsistent with 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 range program of showing that projective determinacy is equivalent to its analytical consequences for the projective sets which would settle positively a conjecture of Woodin and thereby solve the last Delfino problem.