UPPER BOUNDS FOR THE ARITHM ETICAL DEGREES

The study of those degrees which are upper bounds for the arithmetical degrees has been approached from several directions. These degrees were studied by Putnam and his students in an attempt to iterate the jump operator through the arithmetical degrees, a program which was completed by Hodes [4]. Enderton and Putnam [2] showed that if a is an upper bound for the arithmetical degrees, then aC2)>OCw). This led to a study of upper bounds for the arithmetical degrees from a degree-theoretic viewpoint. Results of Sacks [17], Jockusch and Simpson [9], Hodes [5] and Knight, Lachlan and Soare [lo] have added to our knowledge of these degrees. The results of Jockusch and Simpson were instrumental in stimulating further studies of the elementary theory of the poset of Turing degrees. The study of degrees which are upper bounds for the arithmetical degrees has also had interactions with the study of degrees of models of Th(N), i.e., the degrees of models of the elementary theory of the semiring of natural numbers. Feferman [3] showed that the degree of any model of Th(N) is an upper bound for the arithmetical degrees. Harrington, Marker [14], and Knight, Lachlan and Soare [lo] then constructed models of Th(N) with various properties. A characterization of the degrees of models of Th(N) was obtained by Solovay, and improved upon by Marker and Macintyre [13]. Our interest in upper bounds for the arithmetical degrees (ar) was stimulated by the following question of Marker and Knight: Can a minimal upper bound for the arithmetical degrees be the degree of a model of Th(N)? Our investigations led us to study the relationship between other properties of upper bounds for the arithmetical degrees, and analogies between those upper bounds lying below 0’“’ and the degrees below 0 (2) One of our hopes is that the comparison of these sets . of degrees will lead to a definition of 0’ over the elementary theory of the (ordering of the) degrees.