A possible flaw in Theorem 14.17 in Kurt Schütte's -Proof Theory-

Question

Reading Chapter V, 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?

Answer 1

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.