True arithmetic

In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication. The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic. The structure is defined to be a model of Peano arithmetic as follows. The domain of discourse is the set of natural numbers, The symbol 0 is interpreted as the number 0, The function symbols are interpreted as the usual arithmetical operations on , The equality and less-than relation symbols are interpreted as the usual equality and order relation on . This structure is known as the standard model or intended interpretation of first-order arithmetic. A sentence in the language of first-order arithmetic is said to be true in if it is true in the structure just defined. The notation is used to indicate that the sentence is true in True arithmetic is defined to be the set of all sentences in the language of first-order arithmetic that are true in , written Th(). This set is, equivalently, the (complete) theory of the structure . The central result on true arithmetic is the undefinability theorem of Alfred Tarski (1936). It states that the set Th() is not arithmetically definable. This means that there is no formula in the language of first-order arithmetic such that, for every sentence θ in this language, Here is the numeral of the canonical Gödel number of the sentence θ. Post's theorem is a sharper version of the undefinability theorem that shows a relationship between the definability of Th() and the Turing degrees, using the arithmetical hierarchy.