Measure, \(\Pi ^ 0_ 1\)-classes and complete extensions of PA.

*(English)*Zbl 0622.03031
Recursion theory week, Proc. Conf., Oberwolfach/Ger. 1984, Lect. Notes Math. 1141, 245-259 (1985).

[For the entire collection see Zbl 0566.00001.]

Many classical constructions in recursion theory can be viewed as simple forcing arguments. In particular, sets which force their jump are used in many of them and thus they are closely related to the concept of 1- genericity. When studying properties of degrees of complete extensions of Peano arithmetic (PA) the situation is different. In fact, there is a degree of a complete extension of PA which does not bound any 1-generic degree. Constructions using a complete extension of PA as an oracle must, in general, use other arguments and cannot rely on forcing the jump, i.e. on 1-genericity. One of the convenient tools in this case is the use of \(\Pi^ 0_ 1\)-classes. It is the aim of this paper to show properties of \(\Pi^ 0_ 1\)-classes of a special type and by means of these classes to show some results on degrees of complete extensions of PA.

Many classical constructions in recursion theory can be viewed as simple forcing arguments. In particular, sets which force their jump are used in many of them and thus they are closely related to the concept of 1- genericity. When studying properties of degrees of complete extensions of Peano arithmetic (PA) the situation is different. In fact, there is a degree of a complete extension of PA which does not bound any 1-generic degree. Constructions using a complete extension of PA as an oracle must, in general, use other arguments and cannot rely on forcing the jump, i.e. on 1-genericity. One of the convenient tools in this case is the use of \(\Pi^ 0_ 1\)-classes. It is the aim of this paper to show properties of \(\Pi^ 0_ 1\)-classes of a special type and by means of these classes to show some results on degrees of complete extensions of PA.