Reading [Chapter V][1], pages (73-97) in *Proof Theory* (Springer, 1977), by Kurt Schütte, I have encountered a peculiar problem which puzzles me.

On page 96, a map $\rm{Nr}:\overline{\rm{OT}}\rightarrow \mathbb{N}$ is defined using the Cantor pairing function $\pi(m,n)=\frac{1}{2}(m+n)(m+n+1)+m$, where $\rm{OT}$ is the set of *ordinal terms* defined on page 86 and $\overline{\rm{OT}} = \rm{OT}\cup\{\Gamma_{0}\}$. Namely (p.97),

 1. $\rm{Nr}(0) := 1$, $\rm{Nr}(\Gamma_{0})=0$.
 2. $\rm{Nr}(\alpha) = \displaystyle\prod_{i=1}^{n}P_{\pi\left(\rm{Nr}(\alpha_{i})-1,\rm{Nr}(\beta_{i})-1\right)}$, for $\alpha = (\alpha_{1},\beta_{1})\ldots(\alpha_{n},\beta_{n})\in\rm{OT}$ (where $P_{0}=2$ and for $k\ge 1$, $P_{k}$ is the $k$-th odd prime number.

On the same page he provides a definition of the inverse $\tau$ of $\rm{Nr}$, thus proving Theorem 14.17 that $\rm{Nr}$ is a bijection.

All very nice. However, what maps to $31=P_{10}$? Well, $10=\pi(0,4)$, and a simple calculation shows that it should be the term $\alpha = (0,((0,0),0))$. But this is not a principal term, i.e. $\alpha\not\in\rm{OT}$ because it represents $\varepsilon_{0}$, and the unique ordinal term which represents this ordinal is $\beta=((0,0),0)\in\rm{OT}$. But $\rm{Nr}(\beta) = 5$. Then again, what element of $\rm{OT}$ maps to $31$?

What am I not understanding of Schütte's argument?


  [1]: https://drive.google.com/file/d/1x9JpMslhI87jr6Utl4gX7Qf-etGbs6To/view?usp=sharing