Discernible elements in models for Peano arithmetic

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 〈 a 0 , …, a n 〉 we mean a number . Note that n P i 's and a i 's can be nonstandard elements and therefore “finite,” and all concepts which will use it are not absolute but relative to .