The equivalence class containing { | } is labeled 0; in other words, { | } is a form of the surreal number 0. The recursive definition of surreal numbers is completed by defining comparison:  Given numeric forms "x" = { "XL" | "XR" } and "y" = { "YL" | "YR" }, "x" ≤ "y" if and only if: A comparison "y" ≤ "c" between a form "y" and a surreal number "c" is performed by choosing a form "z" from the equivalence class "c" and evaluating "y" ≤ "z"; and likewise for "c" ≤ "x" and for comparison "b" ≤ "c" between two surreal numbers. This group of definitions is recursive, and requires some form of mathematical induction to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via "finite induction" are the dyadic fractions; a wider universe is reachable given some form of transfinite induction. Induction Rule  The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no "Si" with "i" < 0, the expression formula_2 is the empty set; the only subset of the empty set is the empty set, and therefore "S"0 consists of a single surreal form { | } lying in a single equivalence class 0. For every finite ordinal number "n", "Sn" is well-ordered by the ordering induced by the comparison rule on the surreal numbers. The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0≤0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels. For every "i" < "n", since every valid form in "S""i" is also a valid form in "S""n", all of the numbers in "Si" also appear in "Sn" (as supersets of their representation in "Si").