User blog comment:Fejfo/Using functions as fundamental sequences for the fgh/@comment-30004975-20180428220202/@comment-27513631-20180503174033

Yeah, you can check them for equality fine, but you can't check whether two ordinal notations define the same ordinal.

Suppose we could, and let \(\phi(n)\) be a \(\Delta_0\) proposition such that we don't know whether \(\exists n(\phi(n)\). Now, define \(\Phi = \lambda n:\omega.\max\{\texttt{if}\phi(m)\texttt{then}1\texttt{else}0|m<n\}\). \(\Phi = 2\) iff \(\exists n(\phi(n)\) is true, so this would let us decide the consistency of an arbitary theory.

Your idea does have merit, though, and has already been partially explored. Specifically, you've defined the inductive type \(\texttt{NatOrd}\) where \(\texttt{zero}:\texttt{NatOrd}\) and \(\texttt{tree}:(\texttt{Nat} \to \texttt{NatOrd}) \to \texttt{NatOrd}\). You can define a function \(f:\texttt{NatOrd} \to \texttt{Nat} \to \texttt{Nat}\) that defines fundamental sequences.