Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.
Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0.
(1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0.
This is a result of Tennenbaum [T].
(2) There existsd ∈ D0such thatd ≤ 0′.
This follows from the standard Henkin argument.
(3) There existsd ∈ D0such thatd < 0′.
Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.
(4) There existsd ∈ D0such thatd′ = 0′.
Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).
(5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets.
Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.