Example: The Truth Set of Arithmetic
Every arithmetical set is hyperarithmetical, but there are many other hyperarithmetical sets. One example of a hyperarithmetical, nonarithmetical set is the set T of Gödel numbers of formulas of Peano arithmetic that are true in the standard natural numbers . The set T is Turing equivalent to the set, and so is not high in the hyperarithmetical hierarchy, although it is not arithmetically definable by Tarski's indefinability theorem.
Read more about this topic: Hyperarithmetical Theory
Famous quotes containing the words truth, set and/or arithmetic:
“When a man speaks the truth in the spirit of truth, his eye is as clear as the heavens. When he has base ends, and speaks falsely, the eye is muddy and sometimes asquint.”
—Ralph Waldo Emerson (1803–1882)
“Don Pedro. But when shall we set the savage bull’s horns on the sensible Benedick’s head?
Claudio. Yes, and text underneath, “Here dwells Benedick, the married man?””
—William Shakespeare (1564–1616)
“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)