In order to prove these results, Gödel introduced a method now known as Gödel numbering. In this method, each sentence and formal proof in first-order arithmetic is assigned a particular natural number. Gödel shows that many properties of these proofs can be defined within any theory of arithmetic that is strong enough to define the primitive recursive functions. (The contemporary terminology for recursive functions and primitive recursive functions had not yet been established when the paper was published; Gödel used the word "rekursiv" ("recursive") for what are now known as primitive recursive functions.) The method of Gödel numbering has since become common in mathematical logic.