On The Structures Inside Truth-Table Degrees

The following theorems on the structure inside nonrecursive truth-table degrees are established: D egtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is innnite. There are even innnite chains and an-tichains of bounded truth-table degrees inside the truth-table degrees which implies an aarmative answer to a question of Jockusch whether every truth-table degree contains an innnite antichain of many-one degrees. Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and do therefore not contain any objective set. Furthermore a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enu-merable semirecursive sets, one of co-enumerable semirecursive sets and one of sets, which are neither enumerable nor co-enumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integers as for example nineteen, but it is never an even nite number.