In the mathematical field of computability theory, a PA degree is a Turing degree that computes a complete extension of Peano arithmetic (Jockusch 1987). These degrees are closely related to fixed-point-free (DNR) functions, and have been thoroughly investigated in recursion theory. Background In recursion theory, denotes the computable function with index (program) e in some standard numbering of computable functions, and denotes the eth computable function using a set B of natural numbers as an oracle. A set A of natural numbers is Turing reducible to a set B if there is a computable function that, given an oracle for set B, computes the characteristic function χA of the set A. That is, there is an e such that . This relationship is denoted A ≤T B; the relation ≤T is a preorder. Two sets of natural numbers are Turing equivalent if each is Turing reducible to the other. The notation A ≡T B indicates A and B are Turing equivalent. The relation ≡T is an equivalence relation known as Turing equivalence. A Turing degree is a collection of sets of natural numbers, such that any two sets in the collection are Turing equivalent. Equivalently, a Turing degree is an equivalence class of the relation ≡T. The Turing degrees are partially ordered by Turing reducibility. The notation a ≤T b indicates there is a set in degree b that computes a set in degree a. Equivalently, a ≤T b holds if and only if every set in b computes every set in a. A function f from the natural numbers to the natural numbers is said to be diagonally nonrecursive (DNR) if, for all n, (here inequality holds by definition if is undefined). If the range of f is the set {0,1} then f is a DNR2 function. It is known that there are DNR functions that do not compute any DNR2 function. Completions of Peano arithmetic A completion of Peano arithmetic is a set of formulas in the language of Peano arithmetic, such that the set is consistent in first-order logic and such that, for each formula, either that formula or its negation is included in the set. Once a Gödel numbering of the formulas in the language of PA has been fixed, it is possible to identify completions of PA with sets of natural numbers, and thus to speak about the computability of these completions. A Turing degree