Page "Definable real number" Paragraph 12

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.

Since no variables of this language range over the reals, we cannot simply copy the earlier definition of definability.

Rather, we say that a real a 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 φ in the language of arithmetic, with two free variables, such that