Arithmetic Reducibility and Degrees
Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility.
A set is arithmetical (also arithmetic and arithmetically definable) if it is defined by some formula in the language of Peano arithmetic. Equivalently X is arithmetical if X is or for some integer n. A set X is arithmetical in a set Y, denoted, if X is definable a some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in or for some integer n. A synonym for is: X is arithmetically reducible to Y.
The relation is reflexive and transitive, and thus the relation defined by the rule
is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partially ordered under .
Read more about this topic: Arithmetical Hierarchy
Famous quotes containing the words arithmetic and/or degrees:
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)
“The political truths declared in that solemn manner acquire by degrees the character of fundamental maxims of free Government, and as they become incorporated with national sentiment, counteract the impulses of interest and passion.”
—James Madison (1751–1836)