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:
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)
“When a thought of Plato becomes a thought to me,—when a truth that fired the soul of Pindar fires mine, time is no more. When I feel that we two meet in a perception, that our two souls are tinged with the same hue, and do as it were run into one, why should I measure degrees of latitude, why should I count Egyptian years?”
—Ralph Waldo Emerson (1803–1882)