The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.

In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr , r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.