True arithmetic is the set Th of all sentences in the language of first-order arithmetic that are true in . This set is, equivalently, the (complete) theory of the structure (see theories associated with a structure).
Read more about True Arithmetic: Arithmetic Indefinability, Computability Properties, Model-theoretic Properties, True Theory of Second-order Arithmetic
Famous quotes containing the words true and/or arithmetic:
“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous. Rather—speaking loosely and without trying to answer either Pilate’s question or Tarski’s—a version is to be taken to be true when it offends no unyielding beliefs and none of its own precepts.”
—Nelson Goodman (b. 1906)
“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)