  There are two well known systems formalizing total recursion beyond primitive
recursion (\textbf{PR}), system \textbf{T} by G\"odel and system \textbf{F} by
Girard and Reynolds. system \textbf{T} defines recursion on typed objects and
can construct every function of Heyting arithmetic (\textbf{HA}). System
\textbf{F} introduces type variables which can define the recursion of system
\textbf{T}. The result is a system as expressive as second-order Heyting
arithmetic (\textbf{HA}$_{2}$). Though, both are able to express unimaginably
fast growing functions, in some applications a more flexible formalism is
needed. One such application is CERES cut-elimination for schematic
\textbf{LK}-proofs ($CERES_{s}$) where the shape of the recursion is important.
In this paper we introduce a formalism for fast growing functions without a
type theory foundation. The recursion is indexed by ordered sets of natural
numbers. We highlight the relationship between our recursion and the Wainer
hierarchy to provide an comparison to existing systems. We can show that our
formalism expresses the functions expressible using system \textbf{T}. We leave
comparison to system \textbf{F} and beyond to future work.
