Q:

Paul Mahlo's Original Large Cardinal Paper

It is well known that Paul Mahlo (1883-1971) developed a systematic hierarchy of inaccessible cardinals of the type $\pi_{a,b}$ where $\pi_{1,b}$ enumerates the strongly innacessible cardinals, $\pi_{2,b}$ enumerate the fixed points of $\pi_{1,b}$ and so on. My question is, where can I find the original paper of Mahlo in English? If this doesn't exist, are there any good expositional articles on this accessible online?

A:

Just to show what to expect from a machine translation, I spent a little time on two paragraphs of the 1924 Mahlo paper, which I OCR'd and passed to Google Translate. With all the formulas it requires considerable post-processing, but if one is motivated this should be doable in an afternoon.

On a property of a sub-type of the continuum
by Paul Mahlo in Recklinghausen

If one considers a set of subsets of the continuum $C$ of this type, which is so simply ordered that each subset contains all of its preceding real subsets, it can be easily proved that its order type is always a sub-type of the continuum; it therefore contains partial types of regular starting numbers or their
inverses of at most $\omega$ and $^\ast\omega$. In the following we show that such arrangements of subsets of $C$ of the same type can also contain the partial types $\omega_1$ and $^\ast\omega_1$. In doing so we first connect to a final order of the $\omega$-sequences, belonging to continued fractions of positive irrational numbers $x> 1$.
[...]
After these auxiliary considerations, we now deal with a sub-type of the continuum $C$. We denote by $M$ a nowhere dense perfect part of $C$, but always presuppose this $M$ as bounded; all $M$'s therefore have the same type, and every $M$ has $c$ two-sided and $a$ one-sided limits. There exist nowhere dense perfect subsets $M'$ of $C$, that contain $M$ as nowhere dense part, where each one-sided limit of $M$ is a two-sided limit of $M'$. One can easily construct $a$ perfect sets $M_0$, $M_1$, $\ldots$, $M_n$, $\ldots$ ($n <\omega$), so that each $M_n$ contains only two-sided limits of $M_{n + 1}$. The type of elementary sets of all $M_n$ is called $\pi$; it is independent of the particular choice of $M_n$.

