Two theorems on hyperhypersimple sets
Robert W. Robinson · Transactions of the American Mathematical Society · 1967
There are two main results.First, there is a hyperhypersimple set which is not quasimaximal.Second, there is an /--maximal set which is not hyperhypersimple.These results answer questions raised by Young [10, pp.75 and 81] and McLaughlin [3, p. 87].McLaughlin [4] reports a weaker, "non-co-r.e." version of our second result, due to D. A. Martin.He also improves Martin's result, though the"co-r.e." versions of both would be equivalent to our second result.Lachlan [2] has obtained different proofs of the two main results, as discussed in the last section.The proof of the first result involves a construction closely akin to the maximal set construction as handled by Yates [8].The priority system applicable to maximal set constructions was introduced by Friedberg [1], and is essentially unaltered.The proof of the second result requires that the maximal set priorities be altered in an essential way, and thus represents more of a departure.