Combinations of capital letters (e.g., G, EVEN) will be used as names of formulas of Peano arithmetic, x, y, z will be variables. Let be a model for Peano arithmetic. The names of formulas with a bar (e.g., G−, EVEN−(x)) will denote the relations defined by the formulas in . The elements of are called numbers.

By a finite sequence of numbers 〈a0, …, an〉 we mean a number . Note that n Pi's and ai's can be nonstandard elements and therefore “finite,” and all concepts which will use it are not absolute but relative to .