Dedekind cut
Another notion of definability comes from the formal theories of arithmetic, such as Peano arithmetic. The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Because no variables of this language range over the real numbers, a different sort of definability is needed to refer to real numbers. A real number formula_15 is "definable in the language of arithmetic" (or "arithmetical") if its Dedekind cut can be defined as a predicate in that language; that is, if there is a first-order formula formula_31 in the language of arithmetic, with three free variables, such that