Obtaining Woodin's cardinals

Since the 1980's work on Projective Determinacy and AD(L(IR)) the concept of Woodin cardinal has become to be seen as central in the theory of large cardinals and inner model theory. The use by Woodin himself of a background assumption in many arguments that the universe contains unboundedly many such cardinals again draws attention to the centrality of this concept. As is well known the Reflection Principles dating to a more classicial era only provide large cardinals consistent with V = L, and not the wherewithal for such theorems on absoluteness under set forcing that Woodin has proven. We discuss here a reflection principle derived from weak sub-compactness that implies the existence of a proper class of measurable Woodin cardinals thus providing adequate background assumptions for many of Woodin's absoluteness results in his work.