User blog comment:Rgetar/Simple way to create lists of ordinals/@comment-35392788-20181008074754/@comment-35470197-20181009222342

@Rgetar

Right. Converselly, a positive integer \(p_1^{n_1+1} \cdots p_i^{n_i+1}\) corresponds to the finite array \(n_1, \ldots, n_i\), and to \(\beta[n_1] \cdots [n_i]\) as long as \(n_k = 0\) for any \(1 \leq k \leq i\) such that \(\beta[n_1] \cdots [n_{k-1}]\) is a successor ordinal.

Or you can put \(\beta'[n] := \beta'[0]\) for a successor ordinal \(\beta'\). Then the inverse correspondense \(p_1^{n_1+1} \cdots p_i^{n_i+1} \to \beta[n_1] \cdots [n_i]\) gives a total function which is not injective.