 Further, it was a traditional question of descriptive set theory (a question which can be formulated in the language of second order arithmetic) whether all projective sets (see above) are Lebesque measurable. This remained an open problem for many decades, and for a good reason: it turned out that the statement is independent even of the full ZFC set theory (see Solovay 1970). Only by postulating the existence of some extremely large cardinals (so-called Woodin cardinals) can the hypothesis that all projective sets are Lebesque measurable be proved (this was achieved as a consequence of their work on so-called projective determinacy by Woodin, Martin and Steel; see Woodin 1988; Martin &amp; Steel 1988, 1989). 