Where is the (original) proof of Klaus Potthoff's Theorem about the order type of arithmetic models?

Question

I am looking for a complete proof, respectively for the complete original proof of the following theorem, which is attributed to Klaus Potthoff:

If $\mathfrak{M}$ is a nonstandard model of PA, then it cannot have the order type $\mathbb{N} + \mathbb{Z} \cdot \mathbb{R}$.

In the PhD thesis of Bovykin (2000) there is a reference to Smorynski (1984), which refers itself to Pothoff. What I've found so far is a sketch, but I am not sure, if that belongs to the original proof.

Thanks in advance :)

Answer 1

The original proof of Potthoff's theorem can be found in his paper "Über Nichtstandardmodelle der Arithmetik und der rationalen Zahlen" from 1969. Let $\omega + \zeta \cdot \theta$ with $\omega$ the order type of $\mathbb{N}$ and $\zeta$ the order type of $\mathbb{Z}$. Then we have:

Satz 1.2: $\theta$ ist ein offener, dichter Ordnungstyp, der kein stetiges Stück mit mehr als einem Element enthält.

Translation: $\theta$ is an boundlessly and dense order type, which has no continuous part with more than one element. Therefore $\theta$ cannot be the order type of $\mathbb{R}$.

The proof uses Dedekind cuts and therefore differs from the mentioned sketch. If someone is interested I'll translate the original proof from German to English right here.