PA degree

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.

In recursion theory, ${\displaystyle \phi _{e}}$ denotes the computable function with index (program) e in some standard numbering of computable functions, and ${\displaystyle \phi _{e}^{B}}$ 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 χ_A of the set A. That is, there is an e such that ${\displaystyle \chi _{A}=\phi _{e}^{B}}$. This relationship is denoted A ≤_TB; the relation ≤_T is a preorder.

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