Simplicity of recursively enumerable sets

In §1 is given a characterization of strongly hypersimple sets in terms of weak arrays which is in appearance more restrictive than the original definition. §1 also includes a new characterization of hyperhypersimple sets. This one is interesting because in §2 a characterization of dense simple sets is shown which is identical in all but the use of strong arrays instead of weak arrays. Another characterization of hyperhypersimple sets, in terms of descending sequences of sets, is given in §3. Also a theorem showing strongly contrasting behavior for simple sets is presented. In §4 a r-maximal set which is not contained in any maximal set is constructed.