Paul Mahlo's original large cardinal paper

Question

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 inaccessible 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?

Answer 1

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.

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On a property of a sub-type of the continuum

by Paul Mahlo in Recklinghausen

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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 $M_n$.

Answer 2

I have made the articles by Paul Mahlo available at https://www.cs.swan.ac.uk/~csetzer/rareArticles/index.html including bibtex entries. Copyright should have expired by now.

Answer 3

As there is no full answer to this question yet, let me add a partial one which might be helpful:

In a paper of G. H. Müller titled "Reflection in set theory, The Bernays-Levy Axiom System" which is published as part of the collection "Philosophy of Mathematics Today" (edited by Evandro Agazzi and György Darvas), the following passage from Mahlo's 1911 paper is quoted. (See page 158 of the book).

Mahlo, P., Über lineare transfinite Mengen., Leipz. Ber. 63, 187-225 (1911). ZBL42.0090.02.

The text says that the English translation is provided by "D. Reid" but no further reference is added. Also, it is not clear whether it is part of a full translation of the whole paper of Mahlo or just a limited translation for the occasional use in this article. Maybe you can get more information by contacting mentioned people directly.

Therefore we attribute existence to every transfinite number in the definition of which we can find no contradiction if either its equality or its inequality to each otherwise defined transfinite number can be established. This necessary condition is also sufficient, since thus the rank-order between any two arbitrarily chosen transfinite numbers results: otherwise there would have to exist, for a smallest number a, a smallest number $\beta$ for which the size relationship to $\alpha$ could not be established. Were a number $\gamma<\beta$ already greater that $\alpha$, then that would imply that also $\beta>\alpha$, thus establishing the rank-order. Thus $\alpha$ could only be greater than all number $\gamma<\beta$, giving that $\alpha\geq \beta$. If our existence conditions do not imply $\alpha=\beta$, then $\alpha>\beta$, contradicting the assumption that, for $\beta$ as a smallest number, its size relationship to a could not be established. In practice, however, the above condition could remain worthless for a long period, since for example due to our ignorance about certain transfinite numbers, a contradiction could remain hidden in the definition of a proposed new transfinite number. (Transl. by D. Reid.)