Reading [Chapter V](https://drive.google.com/file/d/1x9JpMslhI87jr6Utl4gX7Qf-etGbs6To/view?usp=sharing), 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?



A possible flaw in Theorem 14.17 in Kurt Schütte's -Proof Theory-
Fortunately, there appears to be no flaw in Schütte's construction. Only in my understanding of it.


Ordinal terms are defined on page 86. The term $\alpha=(0,((0,0),0))$ does NOT represent $\varepsilon\_{0}$. After a more careful look at the definitions on page 86 and the definition of $\psi$ on page 84 I see now that $\alpha$ represents


$$\psi0(\psi(\psi00)0) = \psi0\varepsilon\_{0} = \phi0\varepsilon\_{0}^{'} = \omega^{\varepsilon\_{0}^{'}} = \omega^{\varepsilon\_{0}+1} = \varepsilon\_{0}\cdot\omega,$$


where $\alpha'$ denotes the successor of $\alpha$, and $\varepsilon\_{0} = \psi(\psi00)0$, which is the ordinal represented by the term $((0,0),0)$.


Therefore, $\alpha$ is indeed a principal term, properly answering my question "what maps to 31?" The map $\rm{Nr}$ is indeed a bijection.


