  Unfortunately, matters were not to be so simple.  The trouble is that when one climbs the hierarchy of sets in this fashion the greater expressive resources that become available lead to more intractable instances of undecidable sentences and this is true already of the second and third infinite levels.  For example, at the second infinite level one can formulate the statement PM (that all projective sets are Lebesgue measurable) and at the third infinite level one can formulate CH (Cantor's continuum hypothesis).[5]   These statements were intensively investigated during the early era of set theory but little progress was made.  The explanation was ultimately provided by the subsequent independence techniques of Gödel and Cohen.