User blog:Alemagno12/How strong is this function? (2)

Define the set of all ordinal pairs OP using these inductive rules: Next, define an ordinal pair system as a well-ordering on the elements of a subset of OP (which we will call S), where: [WIP]
 * 0 ∈ OP
 * If a,b ∈ OP, then (a,b) ∈ OP
 * If x ∈ S and x is not 0, then x > 0
 * For all (a,b) ∈ S, a,b ∈ S and a,b < (a,b)