Meaning of The Notation
The following meanings can be attached to the notation for the arithmetical hierarchy on formulas.
The subscript in the symbols and indicates the number of alternations of blocks of universal and existential number quantifiers that are used in a formula. Moreover, the outermost block is existential in formulas and universal in formulas.
The superscript in the symbols, and indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type are functions that map the set of objects of type to the natural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers, is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers over numbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects), the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, and so on.
Read more about this topic: Arithmetical Hierarchy
Famous quotes containing the words meaning of the, meaning of and/or meaning:
“Philosophers have actually devoted themselves, in the main, neither to perceiving the world, nor to spinning webs of conceptual theory, but to interpreting the meaning of the civilizations which they have represented, and to attempting the interpretation of whatever minds in the universe, human or divine, they believed to be real.”
—Josiah Royce (1855–1916)
“All the moral laws are readily translated into natural philosophy, for often we have only to restore the primitive meaning of the words by which they are expressed, or to attend to their literal instead of their metaphorical sense. They are already supernatural philosophy.”
—Henry David Thoreau (1817–1862)
“Unlike Freud, Jung did not believe that a dream is a mask for a meaning already known but deceitfully withheld from the conscious mind. In his view, dreams were communication, ideas expressed not always straightforwardly, but in the best way possible within the limits of the medium. Dreaming, in Jung’s psychology, is a constructive process.”
—Jeremy Campbell (b. 1931)